CYBG, owner of Clydesdale and Yorkshire Bank, and Virgin Money, founded almost 25 years ago by British entrepreneur Richard Branson, would combine to create Britain's sixth-largest bank by assets, albeit one still dwarfed by rivals such as Lloyds and Royal Bank of Scotland.

The revised offer values Virgin Money at around 1.6 billion pounds ($2.14 billion) based on Friday's closing share price.

The CYBG-Virgin deal comes at a time when mid-sized banks like them in Britain face competition from both the incumbents with their bigger branch networks and technology budgets, and nimbler digital-only rivals like Monzo, Starling and Atom.

CYBG said on Sunday it had improved its all-share offer for Virgin Money by raising the exchange ratio by 7 percent, an increase which analysts said should be enough to get the deal over the line.

"With Virgin Money management clearly showing less enthusiasm for the fight than we believe is warranted... we suspect that the deal will go through on these revised terms," said Edward Firth, analyst at KBW.

"A clear home-run for CYBG; a reasonable return for Virgin Money shareholders who have had some years of frustration."

Under the terms of CYBG's revised proposal, Virgin Money shareholders would own about 38 percent of the combined group compared with the original 36.5 percent offer.

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While the combined CYBG and Virgin would have assets of around 84 billion pounds, based on the most recent company data, that pales in comparison to rivals like RBS and Lloyds with assets of 739 and 805 billion pounds respectively.

CYBG/Virgin would have around 250 branches, compared with 893 for RBS and 1,795 for Lloyds.

Analysts said that the combined banks could, once merged, go hunting for further acquisitions to achieve scale, in a banking landscape where increasing regulatory and technology costs mean sub-scale players are likely to wither.

Co-operative Bank could be the next target, analyst John Cronin at Irish broker Goodbody said on Monday, as it was rescued last year by hedge funds that are unlikely to be long-term owners and it has an attractive customer base.

Shares in Virgin Money rivals Metro Bank and OneSavingsBank rose on May 8 when Virgin confirmed the CYBG bid, in a sign that investors consider those lenders possible acquisition targets.

CYBG now has until June 18 to make a firm offer.

(Additional reporting by Dasha Afanasieva, editing by Louise Heavens and Adrian Croft)

By Emma Rumney and Lawrence White