The messages dishonestly tell the beneficiary they should tap on a connection for a Covid test as they have been in close contact with somebody who has tried positive.
The messages are sent from cell phone numbers.
Cornwall Council is encouraging individuals to “pause for a moment before clicking any connections”.
Elizabeth Kirk, senior exchanging guidelines official at the chamber, said: “These messages can appear to be exceptionally veritable, however the NHS, government or Public Health Cornwall could never send messages like this from a cell phone.
“The main thing is to recollect that the con artists need you to click a connection.
“Assuming you click the connection, you could empower them to download malware, infections or other nasties.
“It could likewise give them admittance to your passwords for web banking or some other destinations you access from your telephone.”
Cornwall Council said in the event that the connection was tapped on incidentally, the individual should close the site page, reset any passwords, update the telephone’s working framework and square the number the message came from.
They ought to likewise report the episode to Action Fraud and try not to enter passwords or other individual data, it added.
Councilor Martyn Alvey, Cornwall Council’s portfolio holder for public security, said: “Trick calls and messages to cell phones are a not kidding issue and some are very persuading.
“I would ask any individual who gets an instant message that they are uncertain going to reconsider clicking any connections or revealing any private data and look for guidance.”