Spanish bank Santander has submitted an offer to buy a subsidiary of Royal Bank of Scotland, Williams & Glyn's.
RBS is selling the unit and its 318 branches after the European Commission ruled last year that it must dispose of the business as a condition of the bank being bailed out by the British government.
RBS said it was "seriously reviewing" Santander's offer, but added that the tender process remained open to other interested parties.
A number of other banks have been named as potential bidders, including National Australia Bank and Blackrock, a private equity group in the United States.
Virgin Money, which earlier dropped out of the bidding, has told the BBC it was still interested in the RBS branches.
The BBC reports Santander already has a substantial presence in Britain: Abbey National, Alliance & Leicester and Bradford & Bingley.