Stormont's finance committee agreed to monitor the compensation scheme offered by Ulster Bank, on 5 September 2012.
Ulster Bank announced details of a scheme for customers who were affected by a major computer failure over the summer.
Sinn Fein's Mitchel McLaughlin said he wanted the committee to work with the Consumer Council in order to examine other models "by which appropriate response or redress could be brought forward".
"I don't think we should settle to leaving this with the Ulster Bank. Not only do they shroud their processes in mystery, but they are continually making a haimes out of the presentation," he said.
"I would be surprised if people would just simply accept the situation where the bank can determine the response and level of compensation for the inconvenience, and not only this, but the failure of the bank to provide the services for which they were charging customers."
The committee's chairman, Daithi McKay, said representatives from the Consumer Council and Ulster Bank were due to appear again before the committee on 11 October.
The finance committee also heard evidence on the Superannuation Bill.
The Northern Ireland Civil Service (NICS) pension scheme was a statutory pension provision that was established by the Superannuation (Northern Ireland) Order 1972.
If the Superannuation Bill becomes law, the 1972 Order would be amended to remove the need for trade union consent.
Finance department official Grace Nesbitt explained that the bill would also seek to change the compensation scheme for "what people get paid when they leave employment".
She said the changes were necessary to bring Northern Ireland in line with other parts of the UK.