TSB is interested in acquiring the residential loan book of the two building societies, which were nationalised in the run-up to the 2008 financial crisis.
The Government has a 100-per cent shareholding in UKAR Limited, which supports around 467,000 customers with £61.2bn worth of loans.
In June, UKAR said it had begun the potential sale of a portfolio of residential mortgages, but said it would only complete the transaction if the price represented value to the taxpayer.
TSB declined to comment and UKAR was not available for comment.
Reports over the weekend said that the chief executive of TSB, Paul Pester, is interested in buying the residential loan book from UKAR.
TSB’s majority shareholder is Lloyds Banking Group, which is 25 per cent-owned by the Government following a £20bn state bailout in 2008.
In June, Lloyds sold a 35 per cent stake in TSB, valuing the business at £1.3bn.
Lloyds was ordered to sell the 631 TSB branches by European regulators by the end of 2015 as a condition of the bailout, and initially planned to sell one quarter of the business.
Last month UKAR said it could pay off all the money owed to the Government within 10 years.
The bank has already repaid £10.4bn to the Government as rising house prices and low interest rates encourage borrowers to look elsewhere for deals.
Richard Banks, the chief executive of UKAR, which is based in Crossflatts, near Keighley, said: “We expect to pay the majority if not all of the money owed to the Government in the next 10 years – but there are many components and moving parts.
“The interesting fact is that when we’ve repaid the Government debt in circa 10 years’ time, the last mortgage isn’t due to be repaid until 2049.”
UKAR said: “A lot of good work has been achieved to date and we expect to repay the remaining £38.3bn debt in full.”
An 11 per cent reduction in UKAR’s loan book over the 15 months to March 31 was driven by mortgage redemptions, leaving it with 467,000 customers and 529,000 mortgage accounts.