CYBG, the owner of Clydesdale Bank and Yorkshire Bank, must decide whether to make a firm offer or walk away from Virgin by 5 pm on June 18, when a deadline set by the UK's Takeover Panel expires.
It faced an earlier deadline of June 4 but Virgin agreed to an extension after CYBG made an improved all-share proposal earlier this month of 1.2125 new shares for every share in Virgin.
The banks are now on track to reach an agreement before the next deadline expires on Monday, the source said, adding that another extension is not currently expected.
Any deal would be struck on similar terms to CYBG's sweetened proposal, the source said. The value of that bid moves with CYBG's share price but it puts a price-tag of about 1.6 billion pounds on Virgin.
Last minute obstacles could still scupper an agreement.
It comes after CYBG launched its pursuit of Virgin, which is backed by entrepreneur Richard Branson, on May 7 with an initial bid of 1.1297 new shares for every share in Virgin.
Combining the two lenders would create a group better placed to compete with larger rivals such as Royal Bank of Scotland and Lloyds Banking Group, which dominate the British high streets.
Branson owns about 35 percent of Virgin.
(Reporting by Ben Martin, editing by John O'Donnell and Louise Heavens)
By Ben Martin