An 84-year-old from Epsom received a text saying he was a close contact and would need to pay £1.30 for a test kit.
He realised it was a scam before giving any personal information.
However, a 68-year-old from West Sussex ended up transferring £25,000 to a bank account after paying the small charge, then being called by a scammer later.
Any victims should contact police and Action Fraud.