Royal Bank of Scotland, which owns NatWest and Ulster Bank, is extending opening hours for a fifth day but said computer problems have been largely resolved.
A glitch has seen thousands of customers unable to pay in or to receive money into their accounts.
RBS said the problem was fixed over the weekend, but tests were ongoing.
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The director of customer services at RBS, Susan Allen, said she was "cautiously optimistic that RBS and NatWest customer account balances will be largely back to normal from Monday".
"We will automatically waive any overdraft fees or charges on current accounts," she said.
The bank has also promised to liaise with credit agencies to make sure credit scores are unaffected, after many people were unable to pay bills.
"For all other issues, customers have our commitment that they will not be out of pocket from this issue. We will publish further details on how we will ensure this later in the week," said Allen.
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Barclays begins paying up to £100 compensation to customers after banking outageBarclays will pay up to £7.5 million in compensation to customers after its banking services were disrupted by an IT outage
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