McColl's – which has a shop at Morpeth bus station – warned last night that unless it secured more funding, administration was increasingly likely.
A deal with Morrisons would potentially secure 16,000 jobs at the embattled retail chain.
Morrisons is already in a partnership with McColl's, which operates more than 200 Morrisons Daily convenience stores.
The improved offer, made on Thursday evening, is thought to include taking on McColl's pension commitments and its £170m debt.
McColl's is running out of cash and needs an injection of funds to stay afloat.
The retailer has been in discussions with potential lenders to shore up the business, which struggled during the pandemic due to supply chain issues, inflation and a heavy debt burden.
It said that unless those talks are successful, it is "increasingly likely that the group would be placed into administration".
The business has around 16,000 staff, the majority of whom are part-time.
In a statement, in which it described itself as "the UK's leading community retailer", a spokesman said: "As previously disclosed on April 25 2022, the group remains in discussions regarding potential financing solutions for the business to resolve short-term funding issues and create a stable platform for the business going forward.
"However, whilst no decision has yet been made, McColl's confirms that unless an alternative solution can be agreed in the short term, it is increasingly likely that the group would be placed into administration with the objective of achieving a sale of the group to a third-party purchaser and securing the interests of creditors and employees.
"Even if a successful outcome is achieved, it is likely to result in little or no value being attributed to the group's ordinary shares."
Earlier this week, it was revealed the group was set to have its shares suspended from the London Stock Exchange as bosses said they would be unable to get its accounts signed off by auditors in time.
Shares in the company had already plunged as it reported last month that talks with its lenders and banks would likely leave shareholders empty-handed under rescue efforts.