No personal information had been viewed or stolen, the airline said. However, British Airways (BA) has frozen the affected accounts while it resolves the issue.
It means executive club flyers may not be able to use their air miles until the issue is resolved by the airline.
BA said only a small proportion of its millions of customers were affected, and no names, addresses, bank details or other personal information had been accessed.
The airline apologised to customers and said it expected to have the system back up and running in the next day or so, The Mirror reported.
A BA spokesman said: "British Airways has become aware of some unauthorised activity in relation to a small number of frequent flyer executive club accounts. This appears to have been the result of a third party using information obtained elsewhere on the internet, via an automated process, to try to gain access to some accounts.
"We would like to reassure customers that at this stage we are not aware of any access to any subsequent information pages within accounts, including travel histories or payment card details.
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