Bradford & Bingley has approached Northern Rock about buying assets from the troubled lender, the BBC's Business Editor Robert Peston has learnt.
Savers' deposits at Northern Rock are guaranteed by the government
Although the mortgage lender is not looking to buy Northern Rock outright, Mr Peston said it could become part of a private sector rescue of the firm.
The news came after the Treasury extended financial guarantees to Northern Rock at the lender's request.
The government's aid package for the bank now amounts to about £57bn.
It has offered to cover any loss by financial institutions that provide money to Northern Rock so the bank can operate normal banking services.
The Treasury has already guaranteed savers' deposits held at Northern Rock after September saw the UK's first run on a bank in living memory.
Keeping it afloat
The Treasury made the move to ensure that the Northern Rock received enough funding from the money markets to survive into the New Year.
But it means that each taxpayer has a £2,000 exposure to the stricken bank.
The government's previous guarantees were worth £26bn.
According to Mervyn King, the governor of the Bank of England, the additional guarantees amounts to an additional 30% of the Northern Rock's balance sheet.
"This was a natural extension to help the company. What is important now is to find a new management to take the bank forward," Mr King said.
Prime Minister Gordon Brown said on Tuesday that the government's preferred outcome was still to find a buyer for Northern Rock.
"While we will never rule out any option, we will continue the course of action we are pursuing at the moment," he said.
Bradford & Bingley is interested in taking on some of Northern Rock's mortgage assets and could possibly do a deal with either of the consortia looking to rescue the firm.
The auction process has been thrown into doubt recently, though two high-profile suitors remain, including preferred bidder Virgin Group.
"In the worst case of the Rock being nationalised, it could take assets off the Treasury's hands and lessen the very substantial burden and risks for all of us as taxpayers," Robert Peston said of Bradford & Bingley's position.
The Treasury's announcement means that most of Northern Rock's balance sheet is now covered by government guarantees.
The Treasury said it would extend its existing agreement to cover the bank's wholesale obligations and borrowings to other financial institutions.
It said the agreement related to agreements "existing or arising in the future".
But it technically excludes the 40% of Northern Rock's assets that are held offshore in a securitisation programme called Granite, based in Jersey.
The Treasury's move came under fire from opposition MPs, who criticised the use of taxpayers' money to further bailout the bank.
"The government now seems to have got the worst of all possible worlds," said Vince Cable of the Liberal Democrats.
"It's effectively nationalised the liabilities of the bank, while at the same time it doesn't control it."
And the British Bankers Association (BBA) said the possibility of the firm being nationalised was not "a happy proposition".
"Our authorities have dropped the ball and it is not very comfortable at all," said the BBA's Angela Knight.
Northern Rock sought emergency funding from the Bank of England after the global credit markets dried up, news which sent some customers rushing to withdraw their savings.