A senior executive at Ulster Bank's parent company, Royal Bank of Scotland, has said that Ulster Bank customers and those of other banks indirectly affected by the technology problems will be compensated.
Chief Executive of RBS's Corporate Banking Division Chris Sullivan said it was now believed that problems would be resolved for most customers by the week starting 16 July.
However, he told RTÉ's Morning Ireland that this was an informed estimate rather than a certainty.
Customers are continuing to experience difficulties as payments fail to reach their accounts.
The backlog of unprocessed transactions at Ulster Bank stems from a technical breakdown two weeks ago.
Representatives of Ulster Bank are expected to appear before the Oireachtas finance committee today over the bank's technical problems.
Senior members of Ulster Bank are also due at Stormont this morning to try to explain the crisis to a committee of assembly members.
Mr Sullivan said the system breakdown could only be addressed at Ulster Bank once issues had been resolved at RBS and NatWest first.
"Because of that Ulster Bank went past a single day, and then various days became involved in the overall process which caused a corruption of data.
"All of that data had to be manually corrected, and as this was happening, and it took a number of days, transactions were just building up," he added.
Thousands of customers in Northern Ireland are still affected by the computer failure at RBS.
Members of the bank's management team are due to appear before Stormont's Enterprise Trade and Investment Committee at 10am.
Yesterday, Director of Consumer Protection at the Central Bank Bernard Sheridan told an Oireachtas committee the disruption caused to customers was unprecedented and unacceptable.
Mr Sheridan said that the Central Bank would insist on Ulster Bank compensating its customers.
He has also advised that other banks should also be compensated for any costs and charges they had to pay because of the disruption.