Halifax Bank of Scotland, Lloyds TSB and Rabobank have submitted indicative bids to purchase National Irish Bank and Northern Bank.
The three banks have each told the banks' parent, National Australia Bank, that they will only submit a firm bid for the businesses if they are selected as the exclusive bidder.
NAB's chief executive, Mr John Stewart, and his advisers, Lazard, will assess these bids over the coming days and are expected to make a preliminary announcement about the sale process to the Stock Exchange in Australia on Wednesday together with its annual results.
Some sources have suggested the bank could make a more detailed announcement on the future of the two banks on Thursday.
The banks have been valued at more than €1.2 billion.
None of the potential buyers have undertaken a detailed examination of the banks' books. The three financial institutions will also be concerned about the information technology support systems within the two banks and the amount of money they may have to invest in co-ordinating them with their own.
One source described NIB and Northern Bank's systems as "disappointing" and requiring an investment of more than €100 million. NIB and Northern Bank share a common information technology platform that is also shared by their two sister banks in the UK, the Yorkshire and Clydesdale.
NAB has increasingly integrated its four European banks in recent years centralising product development, marketing and other aspects of the business, something which may also take a new owner some time to disentangle.
In terms of staff numbers, the two banks are considered to be relatively lean as they have been restructured and large numbers of employees have left under voluntary severance packages. HBoS is widely regarded as the favourite to enter into exclusive negotiations to buy the banks from NAB.
It currently owns Bank of Scotland Ireland which specialises in providing banking facilities to small and medium-sized business and has a tiny branch network. It operates less than 10 branches in Northern Ireland under the Halifax brand.
It has ambitious plans to move into the retail banking market here and a deal with HBoS would lead to minimal closure of NIB and Northern Bank's existing branches if it were successful in acquiring the two banks.
Lloyds TSB would gain a foothold in the Irish market if it purchased the two banks.
Rabobank, which owns ACCBank and is in discussions about a joint venture with the EBS building society, would significantly expand its presence in the Republic if it emerged as the successful bidder.
