The Forex giant said it powered down its systems as a precaution and to prevent the "software virus" from spreading.
In a statement, the company said its initial investigation "showed no indication any personal or customer data had been compromised".
Travelex has drafted in a team of external cybersecurity experts to "isolate the virus and restore affected systems".
The business will provide foreign exchange services manually at its branches, including those in airports, until the issue is resolved.
It has also issued an apology and apologised for any inconvenience caused.
The incident has had a knock on effect for companies with travel money services provided by Travelex, such as Tesco Bank and Asda.
