RBS is being forced to sell off the branches because of a European Union ruling. The auction is due to end tomorrow, although RBS has until 2013 to find a buyer for the branches.
Santander, which has acquired several hundred Abbey National, Alliance & Leicester and Bradford & Bingley branches last year, is the final bidder after the National Australia Bank dropped out.
According to reports, the Spanish bank is offering between £1.7bn and £1.8bn – well below the £2bn that Santander had initially offered in April.
RBS is 84% owned by the government after public money was used to stop the bank from collapsing at the height of the global financial crisis in 2008.
Santander is currently running an ad campaign backing its Formula One sponsorship of Maclaren and Lewis Hamilton.