The Sir Richard Branson-backed bank, which took control of nationalised Northern Rock in January, was said to have been “very keen” on the business when it was up for sale in 2010.
It is now thought to be considering the opportunity to rebid.
RBS had planned to sell the assets, which feature 316 branches and 40 banking centres for small and medium-sized businesses, to Santander in a deal approved by the European Commission as a condition of receiving state-aid.
The Spanish lender agreed in August 2010 to buy the assets but on Friday night it emerged it had pulled out as it became apparent that a revised target for the purchase to be completed by the end of the year would not be achieved.
The deadline had already been put back twice and it is understood that a review by management consultancy Accenture had indicated it would be mid 2014 before the deal was completed.
Buying the RBS division would more than quadruple Virgin’s branch network and add a small and medium-sized business bank to its offering.
Buyout firm NBNK is also thought to be having a look at the unit.
Gail Cartmail, Unite’s assistant general secretary, urged the Government to press the European Commission to drop its insistence that the branches should be sold.
She said: “This latest development is causing yet more uncertainty and represents another day of chaos for loyal RBS staff.”
Ms Cartmail added: “The real danger is that the European Commission’s requirement to sell branches and assets by the end of 2013 will result in a fire sale and an attempt by any buyer to strip out costs and drive down terms and conditions of hard working staff.”
RBS chief executive Stephen Hester promised there would be “no disruption” for customers and said the taxpayer-backed bank would begin a process to seek a new buyer.
“Much of the heavy lifting associated with a transfer has already been completed, including separating data for 1.8 million customers and putting in place a stand-alone management team.
“It is of course disappointing that Santander decided to pull out of this transaction, especially for the customers and staff involved,” he added.