HSBC has said all of the 275,000 payments held up by a glitch today would be processed by this morning.
The failure yesterday left many workers without their monthly salaries going into the Bank Holiday weekend.
HSBC said last night that 99% of payments that were delayed had now been processed, while those remaining would be credited overnight.
The bank said: "HSBC apologises for the significant inconvenience caused to customers by today's payments problems.
"We are committed to ensuring that no one loses out as a result of today's unacceptable problems. We will work with our customers and the other banks including providing compensation where appropriate.
"Customers who do not receive the delayed payment by Saturday morning should speak with their bank in the first instance. They should also contact their bank if they have suffered any costs as a result of today's problems."
The fault affecting payments relates to the way HSBC transmitted information to the national Bacs payment system. It centred on payments such as invoices and salaries which should have gone out automatically overnight.
While the problem does not relate to payments from banks other than HSBC, their customers will be affected by not receiving money that they expected to have been paid.
HSBC is working with other banks to make sure that, for example, should an account holder not receive their salary as a result of the problem but still makes a regular direct debit payment and goes into an overdraft, he or she would not lose out.
Alan Charlesworth, managing director of insurance and legal recruiter IPS Group, used Twitter to complain: "None of my staff have been paid. HSBC when will this be sorted?"
Another Twitter user wrote: "HSBC. None of my staff have been paid. Cannot get through to them on the phone. Absolute shambles of a bank! Extremely poor performance.
"I trust you are going to reimburse my employees that will incur late payment fees due to your Bacs payments failure."
The glitch comes in the wake of other major banking IT failures affecting customers in recent years, notably by state-backed Royal Bank of Scotland.
It drew criticism from Andrew Tyrie, chairman of the Commons Treasury select committee. He said: "This appears to be yet another serious IT failure by one of the major clearing banks.
"It is increasingly clear that most, if not all, of these banks' IT payments systems need a good deal of investment, and there is a lot to put right."
The Bank of England has been alerted over the glitch while the Financial Conduct Authority (FCA) said it would ensure the bank took steps to help those left at a disadvantage.
Bacs processes transactions worth up to around £50 billion every day and is seen as critical to the functioning of the UK financial system.
Richard Lloyd, executive director of consumer group Which?, said: "Banks have suffered a series of unacceptable failures with their IT systems in recent years and this latest one at HSBC will do little to reassure consumers that banks are making improvements.
"It's essential regulators continue to take tough action to ensure banks properly maintain the payments system we all rely on, and we expect HSBC to fully compensate anyone affected."