A group led by Royal Bank of Scotland has outlined a proposed offer for ABN Amro which values the Dutch bank at 71.1bn euros ($95.6bn; £48.2bn).
It is a rival to the all-share agreed offer from Barclays Bank, which is currently valued at 63bn euros.
The interest from the consortium, which includes Belgium's Fortis and Banco Santander of Spain, has so far been rebuffed by ABN Amro's management.
The offer depends on ABN shareholders rejecting the sale of US unit LaSalle.
Barclays announced it was in exclusive talks with ABN on 20 March
Barclays agreed to merge with ABN in an all-share deal worth 63bn euros on 23 April
Under the terms of the deal Barclays would move its headquarters to Amsterdam
ABN plans to sell LaSalle to Bank of America, but a Dutch court has ruled that it should not do so without consulting its shareholders.
Level playing field
Shareholders' groups said that the sale of LaSalle was designed to prevent RBS making a higher offer for the whole of LaSalle and so was not in their interests.
The 38.40-euros-per-share proposed offer from the RBS consortium is made up of 30.40 euros cash plus 0.844 new RBS shares for each ABN share.
The statement from the consortium also said that its offer would lead to fewer employees losing their jobs than under Barclays' proposals.
RBS-LED CONSORTIUM'S OFFER
The consortium is made up of RBS, Fortis and Banco Santander
The group wrote to ABN on 13 April seeking exploratory talks
The banks also made a $24.5bn offer for ABN's LaSalle unit
On 29 May the consortium outlined a proposed bid which valued ABN at 71.1bn euros
The consortium said that while it would prefer to agree a deal with ABN's management, it would be prepared to make the offer directly to shareholders if necessary.
RBS chief executive Fred Goodwin said he expected the offer to be put alongside Barclays' offer in any shareholder vote.
"All of the bids should go before shareholders, and they should allow shareholders to decide in an environment of having as level a playing field as can be," he said.
RBS shares closed down 0.8% in London trading. Barclays shares were 1.9% higher.
In a letter to staff, Barclays chief executive John Varley said, "There is nothing surprising in today's development and once I have had the opportunity to look at it carefully I'll write to you again."