The Usdaw union said Greybull Capital has lined up accountancy firm KPMG to be administrator for the 120-store business, which employs 1,650 people.
Joanne McGuinness, Usdaw national officer, said she is in talks to try to get staff the best deal.
It is more bad news for sector, coming after BHS and Austin Reid hit trouble.
Greybull's decision to file a notice of intention of appoint administrators gives it protection from creditors for 10 days.
Although Greybull has not commented, it is understood that staff were briefed about the decision on Tuesday.
My Local, formally known as Morrisons' M Local, was sold to Greybull for £25m in a deal fronted by retail veteran Mike Greene, who appeared on Channel 4 show The Secret Millionaire.
The chain has struggled to compete in the cut-throat grocery sector, which is embroiled in a bitter price war.
However, the convenience market remains one of the fastest growing areas of the sector.
There are reports, including by the Daily Telegraph and the Guardian, that Morrisons faces liabilities of up to £20m if My Local goes under.
Ms McGuinness said: "My Local staff are devastated by the news that the company is going into administration. Having been sold by Morrisons last year, there was a mood of optimism that the new owners could turn around the business.
"We are talking with the company in a bid to save jobs and get the best deal for staff. In the meantime Usdaw is providing the support, advice and representation they need at this unsettling time."
The owner of the My Local convenience stores chain, sold by Morrisons last September, may put the retailer into administration.