The business was its convenience store offshoot - known as M Local - before it was sold last September.
My Local has 1,650 staff in some 130 stores. New owner Greybull Capital lined up administrators on Tuesday.
Morrisons said in a statement: "If no buyer is found, and stores close, we will welcome our former colleagues back to a job at Morrisons."
Watch: How many sweet flowers grow in an Easingwold country garden?
Joanita Musisi
Reporter, BBC Radio York
This weekend marks the end of an era for one North Yorkshire man with an amazing garden which he's opened to the public for the last few years to raise thousands of pounds for charity.
The garden, in Easingwold, is the life's work of Henry Dean. But this weekend will be the last time people can see it.
I've been along to see Henry's garden to find out what's in store for visitors in search of floral beauty and tips on how to make their green fingers even greener.
North Yorkshire's weather: Hazy sun and humid with threat of downpours later
We could see more in the way of hazy sun this afternoon but it will start to feel increasingly humid and we're at risk of seeing some heavy, thundery downpours this evening.
A warm afternoon with highs of about 21C (70F),
BBCCopyright: BBC
Headlines: Referendum campaigners in North Yorkshire, events remember Jo Cox MP, Morrisons' My Local offer
Sarah Wakefield
BBC Radio York News
The top stories from the York newsroom this lunchtime include:
Boris Johnson's in North Yorkshire again to campaign for the Leave side in the EU referendum, while former Labour leader Ed Miliband is here pushing for the Remain side
Events are taking place in North Yorkshire and around the world to celebrate the life of Jo Cox - the Yorkshire MP killed last week
World of James Herriot centreCopyright: World of James Herriot centre
With that in mind, here are a few facts about the world's most famous vet:
James Herriot was born on 3 October 1916. His real name was Alf Wight
In 1939, aged 23, he qualified as a veterinary surgeon at Glasgow Veterinary College
In July 1940 he moved to work at 23 Kirkgate in Thirsk (pictured), where he stayed for the rest of his life
He had intended to write about life as a vet for years, but was too busy. In 1966, cajoled by his wife, he started to put pen to paper
He took the pen name James Herriot after seeing the Scottish goalkeeper Jim Herriot, play for Birmingham City against Manchester United
His stories inspired the BBC TV series All Creatures Great and Small, which made household names of the actors Christopher Timothy and Peter Davison
GoogleCopyright: Google
Prison sentence for man who cheated driving theory test in Harrogate
Harrogate Informer
On 13 June 2016, Sarbjit Singh of Regis Heath Road, Rowley Regis, was sentenced to a 32-month prison sentence at Hereford Crown Court for theory test fraud.
Milton Keynes driver theory test centreCopyright: Milton Keynes driver theory test centre
Traffic and Travel update: High Street in Cloughton closed due to an accident
BBC Travel
The A171 High Street in Cloughton is closed and there is very slow traffic in both directions, between the Tattles Hill junction and the Gowland Lane junction, because of an accident.
'We can't import the people we like': Business for Britain on EU referendum
Jonathan Cowap
Presenter, BBC Radio York
Before the polling stations open on Thursday, the two sides in the European Union referendum are making their final push for votes in Yorkshire.
BBCCopyright: BBC
I've been talking to Andrew Lindsay, from Business for Britain, as part of our North Yorkshire referendum debate on BBC Radio York.
Mr Lindsay's based in York and supports the Leave campaign. He's told me that being in the EU makes it "too hard" for people to come to the UK from elsewhere.
Quote Message: Immigration has been very good for Britain. We have a very prosperous economy, but we can't import the people from across the world that we like, we've effectively got to go to Europe first." from Andrew Lindsay Business for Britain
Immigration has been very good for Britain. We have a very prosperous economy, but we can't import the people from across the world that we like, we've effectively got to go to Europe first."
Going to London to visit the Queen: North Yorkshire's MBE honours
BBC Radio York
The Queen is due to honour two people from North Yorkshire today.
Johnny Hayes from York and Emma Woods, from Kirkbymoorside, are on their way to Buckingham Palace to be presented with MBEs
BBCCopyright: BBC
Mr Hayes is receiving his for services to the community on Bishopthorpe Road and Ms Woods' award is for her services to rural business and skills.
Quote Message: I don't know who's presenting the award, but just to be there is amazing and to receive it from any member of the Royal Family would be incredible." from Emma Woods
I don't know who's presenting the award, but just to be there is amazing and to receive it from any member of the Royal Family would be incredible."
Quote Message: I'm just pleased and proud to be given an award for something I care about so much." from Johnny Hayes
I'm just pleased and proud to be given an award for something I care about so much."
Headlines: Final EU referendum push, North Yorkshire tributes to MP Jo Cox and county's MBE honours head for London
Nathan Turvey
BBC Radio York News
The latest headlines from our newsroom in York include:
Both sides in the EU referendum debate are making their final appeals to voters in North Yorkshire before the polls open tomorrow morning
Memorials are being held in North Yorkshire to mark what would have been the 42nd birthday of the Yorkshire MP Jo Cox, who was killed last Thursday.
The Queen is set to honour two people from North Yorkshire who've both been awarded MBEs
Leaving EU would be 'catastrophic': Former PM Tony Blair on BBC Radio York
Jonathan Cowap
Presenter, BBC Radio York
The two sides in the European Union debate are making their final push for votes in Yorkshire ahead of tomorrow's referendum.
BBCCopyright: BBC
I've been talking to the former Labour Prime Minister Tony Blair, who supports the Remain campaign, as part of our North Yorkshire referendum debate on BBC Radio York.
He tells me York and North Yorkshire should vote to stay in the EU, because he thinks the consequences of a leave vote would be economically "catastrophic".
Quote Message: We will end up losing access to the European markets into which we trade half of our goods and services, and vast numbers of jobs depend upon our access to those markets." from Tony Blair Former Labour Prime Minister
We will end up losing access to the European markets into which we trade half of our goods and services, and vast numbers of jobs depend upon our access to those markets."
Watch: Scarborough's fluffy penguin pair get ready to face the world
Adam Tomlinson
I was broadcasting from a beach hut in Scarborough yesterday for BBC Radio York - as you do - and while there I was treated to the sight of this fluffy pair of five-week-old baby penguins.
Beaker and Custard were born to first-time parents at Scarborough Sea Life Sanctuary, but the parents couldn't look after them and so they're having to be hand-reared by staff.
If all goes well, the chicks will join the rest of the flock for the first time in about two months' time.
Celebrations to mark 100 years since birth of North Yorkshire's famous vet James Herriot
Herriot, whose real name was Alf Wight, used his experiences to write several books which inspired BBC TV series All Creatures Great and Small and which brought weekends to a happy, rose-tinted end on Sunday nights for many of us after it began in 1978.
Later this morning, a train named in honour of the world's most famous vet will be re-dedicated at King's Cross Station in London.
Grand CentralCopyright: Grand Central
Quote Message: This is a fabulous opportunity to let Londoners and travellers at Kings Cross know directly more about Thirsk and Herriot Country." from Peter Rush Director of Thirsk Tourist Information.
This is a fabulous opportunity to let Londoners and travellers at Kings Cross know directly more about Thirsk and Herriot Country."
When it comes to music, York is punching well above its weight. No other English town or city of comparable size has anywhere near as many gigs and music events.
North Yorkshire PoliceCopyright: North Yorkshire Police
Ms Pinder was a rear seat passenger in a blue Mercedes which overturned and collided with a tree and metal gate post at about 15:20 on on Thursday 16 June on Kirby Misperton Road.
In a statement, Ms Pinder's family say they are "devastated" at their loss and "still trying to come to terms with it as a family".
The driver of the car, a 27-year-old man from Thornaby, who was arrested in connection with the collision has been released on police bail. The other passengers in the car, a 22-year-old man and an 18-year-old woman, both from Thornaby, have also been released on bail.
Traffic and Travel: 'Serious accident' in Whixley partially blocking B6265
Live Reporting
Andrew Barton
All times stated are UK
Get involved
BBCCopyright: BBC BBCCopyright: BBC BBCCopyright: BBC - Boris Johnson's in North Yorkshire again to campaign for the Leave side in the EU referendum, while former Labour leader Ed Miliband is here pushing for the Remain side
- Events are taking place in North Yorkshire and around the world to celebrate the life of Jo Cox - the Yorkshire MP killed last week
- Yorkshire-based supermarket Morrisons says it'll offer replacement jobs to people working at My Local stores if the shops are closed down
World of James Herriot centreCopyright: World of James Herriot centre - James Herriot was born on 3 October 1916. His real name was Alf Wight
- In 1939, aged 23, he qualified as a veterinary surgeon at Glasgow Veterinary College
- In July 1940 he moved to work at 23 Kirkgate in Thirsk (pictured), where he stayed for the rest of his life
- He had intended to write about life as a vet for years, but was too busy. In 1966, cajoled by his wife, he started to put pen to paper
- He took the pen name James Herriot after seeing the Scottish goalkeeper Jim Herriot, play for Birmingham City against Manchester United
- His stories inspired the BBC TV series All Creatures Great and Small, which made household names of the actors Christopher Timothy and Peter Davison
GoogleCopyright: Google Milton Keynes driver theory test centreCopyright: Milton Keynes driver theory test centre BBCCopyright: BBC BBCCopyright: BBC - Both sides in the EU referendum debate are making their final appeals to voters in North Yorkshire before the polls open tomorrow morning
- Memorials are being held in North Yorkshire to mark what would have been the 42nd birthday of the Yorkshire MP Jo Cox, who was killed last Thursday.
- The Queen is set to honour two people from North Yorkshire who've both been awarded MBEs
BBCCopyright: BBC BBCCopyright: BBC Grand CentralCopyright: Grand Central YorkMixCopyright: YorkMix North Yorkshire PoliceCopyright: North Yorkshire Police GoogleCopyright: Google Brendan CoxCopyright: Brendan Cox
Latest PostGoodbye: Our live coverage across the day
Andrew Barton
BBC Local Live, York
That's the end of our live service for today.
I'll be back from 07:00 on Thursday with more news, sport, travel and weather updates from around the county.
I'll leave you today with a selection of pictures from the stories of the day in North Yorkshire.
Watch: York's Millennium Bridge tribute to MP Jo Cox
BBC Radio York
A special event is due to take place in York later paying tribute to killed Yorkshire MP Jo Cox.
It would have been the Labour MP's 42nd birthday today, but she killed in Birstall last Thursday.
Selena Whitehead has organised the event at York's Millennium Bridge, which is expected to start at about 16:00, and this is what she has to say:
My Local and BHS staff receive job pledges from former owners Morrisons
Yorkshire-based supermarket chain Morrisons has said it will offer jobs to most My Local staff in the event of the chain closing.
The business was its convenience store offshoot - known as M Local - before it was sold last September.
My Local has 1,650 staff in some 130 stores. New owner Greybull Capital lined up administrators on Tuesday.
Morrisons said in a statement: "If no buyer is found, and stores close, we will welcome our former colleagues back to a job at Morrisons."
Watch: How many sweet flowers grow in an Easingwold country garden?
Joanita Musisi
Reporter, BBC Radio York
This weekend marks the end of an era for one North Yorkshire man with an amazing garden which he's opened to the public for the last few years to raise thousands of pounds for charity.
The garden, in Easingwold, is the life's work of Henry Dean. But this weekend will be the last time people can see it.
I've been along to see Henry's garden to find out what's in store for visitors in search of floral beauty and tips on how to make their green fingers even greener.
North Yorkshire's weather: Hazy sun and humid with threat of downpours later
Abbie Dewhurst
Weather Presenter, BBC Look North
Variable cloud will lingering in most places and it will continue to do so for much of the day.
We could see more in the way of hazy sun this afternoon but it will start to feel increasingly humid and we're at risk of seeing some heavy, thundery downpours this evening.
A warm afternoon with highs of about 21C (70F),
Headlines: Referendum campaigners in North Yorkshire, events remember Jo Cox MP, Morrisons' My Local offer
Sarah Wakefield
BBC Radio York News
The top stories from the York newsroom this lunchtime include:
Celebrations mark 100 years since birth of North Yorkshire's famous vet James Herriot
Andrew Barton
BBC Local Live, York
Today marks 100 years since the birth of famous North Yorkshire vet James Herriot, with a train named in his honour being re-dedicated at King's Cross Station in London to mark this special event.
With that in mind, here are a few facts about the world's most famous vet:
Prison sentence for man who cheated driving theory test in Harrogate
Harrogate Informer
On 13 June 2016, Sarbjit Singh of Regis Heath Road, Rowley Regis, was sentenced to a 32-month prison sentence at Hereford Crown Court for theory test fraud.
Traffic and Travel update: High Street in Cloughton closed due to an accident
BBC Travel
The A171 High Street in Cloughton is closed and there is very slow traffic in both directions, between the Tattles Hill junction and the Gowland Lane junction, because of an accident.
'We can't import the people we like': Business for Britain on EU referendum
Jonathan Cowap
Presenter, BBC Radio York
Before the polling stations open on Thursday, the two sides in the European Union referendum are making their final push for votes in Yorkshire.
I've been talking to Andrew Lindsay, from Business for Britain, as part of our North Yorkshire referendum debate on BBC Radio York.
Mr Lindsay's based in York and supports the Leave campaign. He's told me that being in the EU makes it "too hard" for people to come to the UK from elsewhere.
Going to London to visit the Queen: North Yorkshire's MBE honours
BBC Radio York
The Queen is due to honour two people from North Yorkshire today.
Johnny Hayes from York and Emma Woods, from Kirkbymoorside, are on their way to Buckingham Palace to be presented with MBEs
Mr Hayes is receiving his for services to the community on Bishopthorpe Road and Ms Woods' award is for her services to rural business and skills.
Headlines: Final EU referendum push, North Yorkshire tributes to MP Jo Cox and county's MBE honours head for London
Nathan Turvey
BBC Radio York News
The latest headlines from our newsroom in York include:
Durham v Yorkshire - Day four
Follow live text and radio commentary of Durham v Yorkshire in the County Championship.
Catch upLeaving EU would be 'catastrophic': Former PM Tony Blair on BBC Radio York
Jonathan Cowap
Presenter, BBC Radio York
The two sides in the European Union debate are making their final push for votes in Yorkshire ahead of tomorrow's referendum.
I've been talking to the former Labour Prime Minister Tony Blair, who supports the Remain campaign, as part of our North Yorkshire referendum debate on BBC Radio York.
He tells me York and North Yorkshire should vote to stay in the EU, because he thinks the consequences of a leave vote would be economically "catastrophic".
Watch: Scarborough's fluffy penguin pair get ready to face the world
Adam Tomlinson
I was broadcasting from a beach hut in Scarborough yesterday for BBC Radio York - as you do - and while there I was treated to the sight of this fluffy pair of five-week-old baby penguins.
Beaker and Custard were born to first-time parents at Scarborough Sea Life Sanctuary, but the parents couldn't look after them and so they're having to be hand-reared by staff.
If all goes well, the chicks will join the rest of the flock for the first time in about two months' time.
Celebrations to mark 100 years since birth of North Yorkshire's famous vet James Herriot
Andrew Barton
BBC Local Live, York
A ceremony's due to take place later to mark 100 years since the birth of famous North Yorkshire vet James Herriot.
Herriot, whose real name was Alf Wight, used his experiences to write several books which inspired BBC TV series All Creatures Great and Small and which brought weekends to a happy, rose-tinted end on Sunday nights for many of us after it began in 1978.
Later this morning, a train named in honour of the world's most famous vet will be re-dedicated at King's Cross Station in London.
York open mic nights: Your complete guide
YorkMix
When it comes to music, York is punching well above its weight. No other English town or city of comparable size has anywhere near as many gigs and music events.
Our listing is organised by day (weekly events first, then the monthlies).
Some organisers ask performers to book a time to play, but for most events it is just "turn up and hope to get a slot".
Thornaby teenager named as Kirby Misperton crash victim
A 19-year-old woman who died in a crash in North Yorkshire last week has been named as Lucy Pinder, from Thornaby, Stockton on Tees.
Ms Pinder was a rear seat passenger in a blue Mercedes which overturned and collided with a tree and metal gate post at about 15:20 on on Thursday 16 June on Kirby Misperton Road.
In a statement, Ms Pinder's family say they are "devastated" at their loss and "still trying to come to terms with it as a family".
The driver of the car, a 27-year-old man from Thornaby, who was arrested in connection with the collision has been released on police bail. The other passengers in the car, a 22-year-old man and an 18-year-old woman, both from Thornaby, have also been released on bail.
Traffic and Travel: 'Serious accident' in Whixley partially blocking B6265
BBC Travel
There's been an accident at Score Ray Lane in Whixley on the B6265 which has partially blocked the road in both directions.
The accident, involving a car and a motorbike, is described as "serious." Police are in the area directing traffic.
MP Jo Cox: Birthday tribute planned at York's Millennium Bridge
BBC Radio York
An event is due to take place in York later to mark what would have been the 42nd birthday of killed Yorkshire MP Jo Cox.
The Labour MP for Batley and Spen was shot and stabbed to death in her constituency, near Leeds, last week.
The gathering at York's Millennium Bridge this afternoon is one of several events taking place all over the world to celebrate Mrs Cox's life.