The bank is hoping to avoid a takeover by the Irish government
The finance workers' union has said staff at First Trust have been left shocked by the decision to sell it off.
Larry Broderick, the Irish Bank Officials Association (IBOA) general secretary said members were "profoundly distressed, confused and angry".
First Trust's parent company Allied Irish Banks (AIB) has confirmed that the UK arm is up for sale as part of its efforts to raise 7.4bn euros.
AIB is selling its divisions in the US and Europe, including First Trust.
The bank employs about 1,300 people in 48 branches across Northern Ireland.
AIB's decision to sell First Trust and its banking arm in Britain comes after the bank sold 23bn euros worth of loans to the Republic's National Asset Management Agency (Nama), formed to take control of toxic assets, at an average discount of 43%.
As the loans were sold at a discount, the bank now has a "black hole" in its funding.
If it fails to raise further private investment, the Irish state will take a majority shareholding in the bank.
In the clean-up operation of Irish banking, Nama will take more than 1,200 individual toxic loans off the books of five lenders - Bank of Ireland, AIB, Anglo, Irish Nationwide and EBS.
Mr Broderick said the union had been flooded with calls from members over the decision to sell First Trust.
"Our members in First Trust Bank who have worked so hard to build up the business in the most difficult circumstances feel betrayed by senior management team that considers them to be dispensable."
Mr Broderick said the union is seeking urgent meetings with AIB's senior management.
On Tuesday, another bank, Anglo Irish, announced the largest corporate loss in the history of the Republic of Ireland.
It made a loss of 12.7bn euros ($17.2bn; £11.4bn) in the 15 months to December.
The announcement came a day after the Irish government said it would inject a further 8.3bn euros into the bank.
Irish Finance Minister Brian Lenihan said pumping in more money into the bank was "the least worst option".