The Musgrave group says it has been the subject of a cybercrime attack, during which hackers tried to steal debit and credit card numbers and expiry dates.
But the owner of the SuperValu, Centra and Daybreak chains says there is no evidence that any data has been stolen so far.
The attempt was discovered yesterday afternoon when malicious software was discovered in a centralised IT system.
The company says the system did not store cardholder names, PIN numbers or CCV numbers and as a result these could not have been compromised.
Musgrave says it is advising customers to review activity on their statements as a precautionary measure and has apologised for the issue.
Gardaí and the Office of the Data Protection Commissioner have been notified of the attempted breach.
In a statement, the firm said its cyber breach response experts have installed "advanced technical fixes and continue to actively manage and monitor the situation."
It said a range of security solutions including threat-monitoring, anti-virus software, firewall and penetration testing are all deployed.