The messages where the sender is pretending to be from the official NHS bodies were reported by the health boards on Friday.
Some messages appear to ask the receiver to click on an unverified link to order a test kit. They claim that the individual has been in contact with someone with the virus.
The link then asks individuals to enter their bank details.
The official NHS boards have urged people not to open these links.
An NHS Shetland spokesperson said: “We’ve been made aware of various scam texts circulating Shetland regarding being a close contact.
"We’ve attached two of the scam texts below.
"Please do not open these links or the messages.
"As a reminder, if you are a close contact, you will receive a message from “Test and Protect".
It is understood that certain messages specify that a person has been in contact with someone who has the Omicron variant of the virus.
An NHS Western Isles spokesperson said: “We are aware of the following message which a number of people have reported that they received.
"This is not a genuine message, please do not respond.
"The NHS will not ask you to order a test kit at a cost.
"Please do not provide bank details to sources you are unsure of.”