Ulster Bank's Chief Operating Officer Ellvena Graham has unreservedly apologised to all those affected by its IT problems.
Ms Graham said the bank would ensure that no customer of Ulster Bank or any other bank would be out of pocket and would assess claims on a case-by-case basis.
The bank says the issue has now been fixed but it will take the weekend to clear the backlog, which amounts to millions of euro in transactions.
The bank was criticised this morning for its silence and approach to the problem by Ministers Richard Bruton and Joan Burton.
Ms Graham said the problem had arisen within the systems of parent bank RBS when an incorrect patch was applied.
She said they are urging customers to go into branches and any issue will be dealt with.
She said they have also talked to other banks and anyone who has not received money due via Ulster Bank, their own bank should make good on it.
Ms Graham also said that bank branches would open on Sunday if necessary.
Mr Bruton said the bank were "remarkably silent" on the problem.
He said Ulster bank must put the protection of its customers to the fore and the bank should give them a explanation as to what has caused the problem.
The Minister said 100,000 customers are getting frustrated and are really worried about their situation, and whether their credit rating will be affected or if they will face additional charges.
He said the bank must get this right and make sure it protects its customers in the course of sorting it out.
Ulster Bank will keep 80 branches open until 7pm today and 30 branches will open on Sunday as it attempts to deal with the problems caused by the technical fault.
The fault prevented pay cheques and social welfare payments being credited to accounts over the past 48 hours.
The bank said that ATM and Anytime Internet and Mobile Banking services are available but are reflecting a delay in customers' balances.
The HSE has made arrangements for staff to be paid tomorrow.
In the case of those that require access to funds before that they can withdraw up to €500 from their own bank with identification and a HSE payslip.
The withdrawal can be up to €1,000 at the discretion of the bank manager in some cases.
Staff whose accounts are with Ulster Bank, can seek funds in this way or, as things stand, will have to wait until Monday to access funds.