The aviation regulator said it had taken "urgent action" over the suspicious online activity and will inform the police.
Some of the concern is understood to have risen from a series of low-level claims, which often indicates fraud.
The website was set up to give refunds to people with Atol-protected bookings for Thomas Cook holidays due to begin after the 178-year-old firm collapsed on September 23.
This relates to more than 360,000 bookings covering trips set to be taken by 800,000 people.
Thousands of customers passengers were repatriated in a huge operation to bring stranded customers from overseas following the collapse.
Atol-protected customers who were already abroad when Thomas Cook failed can also claim for the cost of replacing the parts of their holiday which were financially protected, or out-of-pocket expenses for delayed flights.
The CAA is aiming to pay refunds within 60 days of receiving a valid form and wants to crack down on fraudulent activity to avoid any delay.
Civil Aviation Authority chair Dame Deirdre Hutton said: "This morning we have taken urgent action in response to what we believe is attempted fraudulent activity in relation to refunds for Thomas Cook customers.
"If you have made a claim directly with us, then your claim is being processed and you do not need to take any action."
She added: "Please help us to combat the risk of fraud by not submitting your details to any other website.
"Our focus is on getting money back to the right people as soon as possible and combating fraud in every way possible."
It is the latest setback for customers seeking their money back after the CAA was forced to apologise on Monday that its system struggled to cope with "unprecedented demand" in the hours after it launched.
Many people received an error message after entering their details, meaning their claims were not submitted.
As well as the hundreds of thousands holidaymakers impacted by the collapse, some 9,000 staff in the UK were left jobless when the business failed to secure a last-ditch rescue deal.
Thomas Cook failed to obtain rescue funds from its banks and was left with debts totaling more than £1 billion.