Santander UK has put forward a bid to buy what would be RBS' Williams & Glyn unit.
RBS was instructed to sell the Williams & Glyn unit, which is more than 300 branches strong, by the end of 2017, as part of its £45bn state bailout deal. However, the bank warned in April that it might not make this deadline.
Sources also told Sky News, which first reported the news, the deal was still tentative and it was not certain it would go ahead. The deal is also unlikely to be formalised by the end of the week.
RBS is due to report half-year results on Friday, where many investors will likely be anxious to hear about the progress being made on the sale of the unit.
Shares in RBS spiked on the news, but are down 1.3 per cent today. The bank declined to comment.
A statement from Santander read: "While our focus is organic growth, we will continue to analyse opportunities in our core 10 markets where they add value and benefit to our customers and shareholders.
"That said, we do not comment on rumours or market speculation."
This is not the first time Spanish lender Santander has expressed an interest in RBS' branches, with a failed attempt in 2012, nicknamed Project Rainbow, falling through due to IT issues.
The news comes less than a week after Lloyds Banking Group revealed it would be closing the doors on a further 200 of its branches, thanks to a decline in footfall and concerns that a further cut in interest rates could place further strain on the bank's financials.