Sky News sources said Santander UK is interested in securing Williams & Glyn through an asset transfer which would result in its loans, deposits, branches and around 3,000 customers, transferred to Santander.
RBS and Santander are reported to be unlikely to reach an agreement on the transfer of assets this week, but the Treasury has allegedly been informed about the offer.
The government made the sale of the 314 branches a condition of the RBS bailout package in 2008 when it stepped in following the financial crisis to save the bank from collapse.
Williams & Glyn made an application for a banking licence in October 2015 and RBS had planned to launch an Initial Public Offering at the end of 2016.
Selling assets rather than creating a new fully-formed bank would, according to Sky, yield a lower price for RBS but it would allow the bank to offload the branches quickly.
RBS will report on its half year results on Friday and is expected to provide an update on its Williams & Glyn plans at that time. Sky sources said that the deal still remained uncertain and could fall apart.
A spokesperson from Santander said: “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.”