The scam has seen residents receive a text message which includes a link to a fake NHS website.
They are then asked to submit their personal details and pay a 99p postage cost for the test.
Later the same day, the victims receive calls from a number claiming to be from a fraud department, asking them to download an app.
Once this is done, they give the scammer full access to their laptop and online banking, meaning residents have been duped out of thousands of pounds.
Detective Sergeant Sam Erabadda said: “We know that thankfully, many people will treat text messages like this with suspicion, but with their know being a cost associated with Covid-19 tests, it’s easy to fall for a scam like this which ask for payment.
“The scammers are professionals so the messages and webpages can be quite convincing, and sophisticated criminals often disguise their telephone number so even if Googled, they will appear to be from a bank or other reputable organisation.
“If you receive a scam text message which you think might be suspicious, do not click on any links in the message, and do not make payment, provide your personal details or install any apps if prompted.
“You should forward the message to 7726, a free spam-reporting service provided by phone operators, before deleting the message.
“I’d really encourage everyone to spread the word about scams like this, and ensure their loved ones know to be vigilant.”