The attack appears to affect tens of thousands of customers and some cards have been blocked.
The bank told the BBC that some accounts "have been subject to online criminal activity, in some cases resulting in money being withdrawn fraudulently."
Tesco Bank CEO Benny Higgins told Radio 4's "Today" programme that 40,000 accounts had suspicious transactions.
He said the amount stolen was a "big number but not a huge number."
BBC personal finance correspondent Simon Gompertz said money has been taken from 20,000 accounts.
Money taken from 20,000 accounts in Tesco Bank fraud. Tesco boss promises refunds
- Simon Gompertz (@gompertz) November 7, 2016
As a precaution we have notified some customers that we have blocked their cards to protect their account - https://t.co/m8zjO6BKgj
- Tesco Bank Help (@tescobankhelp) November 6, 2016
Higgins said that any financial loss incurred by customers will be resolved fully by Tesco Bank.
Tesco Bank did not immediately respond to Business Insider's request for comment.
This is a developing story. More to follow...