Challenger bank CYBG, formerly known as Clydesdale and Yorkshire Banking Group, has increased its offer for Virgin Money.
The new offer actually has a lower monetary value than the initial offer made last month.
This is because the proposed acquisition would be entirely paid for through CYBG shares, and the shares have fallen in value since last month, when the initial offer was made.
The new proposal seeks to compensate for this by offering Virgin Money shareholders 38 per cent of the enlarged bank, compared with the previous offer's 36.5 per cent.
The boards of both companies are in discussions regarding the new offer, while emphasising that there is no certainty a formal offer will be made.
The board of Virgin Money said it was entering discussions about the offer because of the enhanced potential for synergies between the companies, and the increased opportunity for its shareholders to benefit from the growth of the combined business.
Based on CYBG's share price on 1 May, the offer values Virgin Money at about £1.55bn.
The joint statement to the stock market from both companies stated: "The Boards of CYBG and Virgin Money believe that the proposed combination would create the UK's first true national banking competitor, offering both personal and SME customers an enhanced alternative to the large incumbent banks.
"The proposed combination would provide a powerful full-service banking offer for around six million personal and business customers, bringing together the complementary strengths of CYBG and Virgin Money."