According to the website Downdetector, which tracks online outages, there have been thousands of reports of faults with TSB since 11:45am on Tuesday.
The worst affected areas of the UK appear to be London, Glasgow, Edinburgh, Birmingham and Aberdeen.
In a statement on social media, TSB told customers: “We're aware some customers are currently experiencing problems across our mobile app and Internet Banking.
“We're working to solve this as soon as possible and apologise for any inconvenience caused.”
Some customers have complained that, since their local branches closed, the latest IT issue has left them without access to their money.
One Twitter user complained to the bank: “Not only has the mobile app and internet banking now not been working since this morning, but because you have closed my local branch, customers are completely isolated.”
TSB has been approached for comment.