The HMRC has set up a helpline for concerned individuals and Apacs is warning people to be vigilant and check their bank statements.
Apacs says there is no evidence the data has fallen into the wrong hands but if it does, individuals will be protected by the Banking Code.
It adds bank details, national insurance numbers, date of birth and names and addresses is insufficient for fraudsters to access a bank account as they will need additional security information and passwords.
Apacs chief executive Paul Smee says: “While this incident is extremely serious, at this stage customers should not be unduly concerned, as there’s no evidence that the data has fallen into criminal hands.
“As part of its usual procedures the banking industry has done all it can to protect its customers accounts and will continue to do so. In the event that anyone is the innocent victim of fraud as a result of this incident customers can have peace of mind that they enjoy protection under the Banking Code which means that you should not suffer any financial loss as a result.”