The Financial Services Union has said uncertainty is growing over the future of Ulster Bank's operations in the Republic, following a meeting today between union officials and the chief executive of the bank, Jane Howard.
The FSU described the meeting as "difficult and deeply troubling" and said it provided no reassurance for staff or customers of the bank.
Ulster Bank's parent, NatWest, has confirmed that it is undertaking a strategic review of its operations in the Republic of Ireland, but that no decision had yet been taken.
There is speculation that the bank may announce the outcome of the review when it publishes its annual results on Friday of next week.
"The CEO of Ulster Bank refused to answer any questions about the strategic review or to adequately respond to recent media reports about possible talks between NatWest and PTSB," said John O'Connell, General Secretary of the Financial Services Union.
"The union left the CEO under no doubt as to the level of anxiety and stress that bank staff are under," he added.
Ulster Bank employs 2,400 people here across 88 branches.
Mr O'Connell said he believed the threat of a closure of carve-up of the bank is growing.
"This will seriously impact communities, customers, staff, and businesses all over Ireland," he said.
He said the manner in which people have been treated is "totally unacceptable".
"This union will not stand by while loyal staff are treated so poorly.
"Their loyalty, commitment and well-being need to be central to the review process," he said.
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According to the FSU, staff are being asked to undertake work in respect of the process that could enable the sale or carve-up of Ulster Bank.
"The bank is asking them to sign non-disclosure agreements, so they cannot discuss it," said Mr O'Connor.
If NatWest decides to exit the Irish market, Mr O'Connell said jobs and customers need to be protected.
"The FSU clearly stated in the meeting that a carve up of the bank's assets is not an acceptable outcome and under no circumstances should a vulture fund be able to get its hands on any part of the loan book," he said.
The FSU said it is calling on the Minister for Finance and the Central Bank of Ireland to make urgent statements about the situation.
"A decision is coming, and it is coming fast – action is therefore needed now," said Mr O'Connell.
On Wednesday, Ulster Bank staff who attended an online meeting with the Financial Services Union expressed their "fury and dismay" at the lack of communication between the bank and its employees over its future, according to the FSU.