Finance Union IBOA has requested a meeting with UK Chancellor of the Exchequer George Osborne about the future of Ulster Bank.
IBOA General Secretary Larry Broderick said the union wants greater clarity on the future of the bank after Mr Osborne raised the possibility of removing bad assets from Ulster Bank and its parent Royal Bank of Scotland.
The IBOA said it also wants to talk to Minister for Finance Michael Noonan and Ulster Bank Chief Executive Jim Brown.
The proposal, in a recent British treasury position paper, is to offload Ulster Bank to the State in exchange for some NAMA properties located in the UK.
Mr Broderick said: "Such a development would have major implications for competition within Irish banking, and as such, would be of concern to both the customers and the employees of all banks operating on the island of Ireland."
Mr Osborne raised the possibility of Ulster Bank's assets being sold off from its parent RBS last evening during a speech in London.
He also said that he would like the bank to separate out toxic assets and risky loans from parts of the business that support the British economy.
The review will particularly focus on assets in Ulster Bank and UK commercial real estate, and will not involve any further injection of taxpayer money into RBS.
Speculation has been mounting that the British treasury wants to begin the process of selling its stake before the 2015 general election.
British Prime Minister David Cameron recently raised the prospect of selling RBS shares at a loss.
In February, Ulster Bank reported losses of more than €1.2 billion for 2012, which was a slight increase on the previous year.
Ulster Bank has a strong presence on both sides of the Irish border, with a network of 137 brances in the Republic and 86 in Northern Ireland.
Its mortgage arrears are at 17%, which is 5% higher than average.