It came after the bank, which has five million customers, shut down online banking and limited some other services over the weekend for planned work.
The City regulator and the Information Commissioner’s Office (ICO) have both been in contact with the lender over the issue.
Over the weekend TSB, now owned by Spain’s Sabadell banking group, was migrating all its customers onto a new banking platform. It had previously been using systems belonging to its former owner Lloyds Banking Group.
It said this migration occurred correctly. However, problems began to arise when the system was switched back on again on Sunday evening, prompting customers to raise concerns on social media.
One said he could see other people’s accounts totalling more than £20,000, while another reportedly discovered he had been wrongly credited with £13,000 after logging back in.
Account holders had been warned last week that services faced disruption from 4pm on Friday as IT work was carried out, with systems turned back on again at 6pm on Sunday.
It was then that “access issues” affecting what TSB called a “tiny fraction of our customer base” occurred for about 20 minutes, the bank said, before being fixed.
The bank said these related to nominee accounts, where certain individuals such as parents or children are able to access another person’s account in branch or on the phone.
TSB’s glitch on Sunday meant that for 20 minutes these people were also able to access accounts online, it said.
A spokeswoman said that 2% of its five million customers have such accounts.
Services were shut down for several hours after the problem was discovered and resumed at around 2am, TSB said.
It blamed Monday’s “intermittent issues” with account access on the fact that large volumes of people were logging on again having been unable to do so over the weekend.
The bank said: “We are really sorry for the inconvenience this is causing our customers and want them to know we are working as hard and as fast as we can to resolve this problem.”
The Financial Conduct Authority said it was aware of the disruption and was in touch with bank officials.
A spokesperson for the ICO said: “We are aware of a potential data breach in relation to the TSB and are making enquiries.”
One Twitter user, Tracy Hannah, wrote: “@TSB as soon as I can get access to my own money I’m closing my accounts down. Not been able to get money since Friday but you can still access payments, tsb is the worst bank I’ve ever dealt with. #tsb.”
Others used more colourful language to describe their frustration.
Among those customers contacting TSB’s social media included several who said they could see other people’s accounts after logging back in.
Craig Malcom tweeted: “@TSB I currently have access to £20k+ of other peoples money.
“I suggest somebody answers the phones as iv been on hold for 45 minuets!
“This is a MASSIVE breach of data protection! If i have access to their account they could have to mine as well!”
Another Twitter user called Bex said: “@TSB so go to my app and have someone else’s accounts there!!!! Serious dpa (Data Protection Act) breach! Want to speak to someone now and a half an hour wait!! What would the FCA (Financial Conduct Authority) say about this!!! Might just inform them a bank is giving away other people’s account numbers”.