Santander has made a formal offer to buy Williams & Glyn from RBS, according to Sky News.
Santander is looking to transfer over all the Williams & Glyn assets, including its 314 branches, 3,000 customers, loan book and deposit book.
The Treasury has reportedly been told about the offer.
The Government is forcing RBS to sell Williams & Glyn as part of its 2008 bailout terms.
Sky News says that RBS selling assets rather than creating a new bank speed up the deal but yield a lower price to the seller.
RBS will report on its half year results on Friday.
A Santander spokeswoman says: “Whilst our focus is organic growth, we will continue to analyse opportunities in our 10 core markets where they add value and benefit to our customers and shareholders. That said, we do not comment on rumours or market speculation.”
An RBS spokesman would not comment.