Some Tesco Bank customers are still locked out of their online accounts a week after the bank moved 605,000 accounts to a new computer system.
Chief executive Benny Higgins admitted they had failed some customers.
The BBC has also been receiving complaints about the time taken to get through to customer service helpline staff.
Mr Higgins apologised "unreservedly", but admitted that 2,500 customers could not access their accounts.
"Tesco strive to deliver service of the highest standard and for a significant minority of customers we have failed to do that," Mr Higgins told BBC Radio 4's Money Box.
"We do apologise unreservedly. It's absolutely our focus to put this right."
The problems began on Monday after Tesco Bank migrated all its online savings accounts from the Royal Bank of Scotland system to its own computers.
Many customers have been contacting news organisations and posting messages on web forums complaining that the new system was rejecting them, and that they were unable to access their funds.
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Twenty-five thousand customers had already registered under the new system, and 9,000 were actively managing their account, Mr Higgins told the programme.
But he confirmed that two technical failures on Monday and Tuesday lasting 18 hours had left people who were online at the time unable to access their accounts until they took several steps to reset their security details.
"We had 2,500 customers locked out as a result of Monday and Tuesday and we have been contacting them by telephone and e-mail."