The bank, a subsidiary of British supermarket giant Tesco, the kingdom's biggest retailer, said it was trying to refund accounts as quickly as possible.
"Tesco Bank can confirm that, over the weekend, some of its customer current accounts have been subject to online criminal activity, in some cases resulting in money being withdrawn fraudulently," chief executive Benny Higgins said in a statement.
The bank confirmed that of its 136,000 current account holders, 40,000 had seen suspicious transactions over the weekend, while money had been fraudulently withdrawn from around 20,000 accounts.
Tesco shares dipped by 1.28 percent to 199.90 pence in early London trading, as London stocks rose by 1.3 percent.
"We can reassure customers that any financial loss as a result of this activity will be resolved fully by Tesco Bank," said Higgins.
The retail bank is working with the National Crime Agency and the Financial Conduct Authority to address the security breach.
"It will be investigated and hopefully that will lead to action and arrests," an NCA spokesman said.
The spokesman said cyber-attacks tended to vary in terms of sophistication so there was no set formula for dealing with them.
Tesco Bank opened in 1997 and has 7.8 million customer accounts.
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