The Ulster Bank says it is not in a position yet to give an exact day or time for a return to full service.
The organisation says it is doing all it can to bring an end to the crisis which is currently in its second week.
Around a third of the bank's branches in Northern Ireland opened on Saturday to try and clear the huge backlog of transactions.
Stephen Cruise, of Ulster Bank, says they will continue to give updates.
"The important thing is, we've our staff working round the clock and this weekend is very important to us to really accelerate and catch up in the backlog."
He said that customers should continue to call the helpline number.
"We've ten branches open tomorrow, an additional two - one in Andersonstown, one in Connswater - that weren't open last Sunday from 10am through to 1pm.
"Please come and talk to us and we will assist you to get out your cash or make payments or whatever it is that you need to do."