Health experts are focusing on two areas of the Dudley borough as a precautionary measure after three cases of the Indian variant were identified.
Dudley Council bosses said they are tracking a "very small number" of cases of the B16172 Indian variant in Halesowen and Brierley Hill.
People in these areas who have symptoms or have been in contact with someone with a positive test are being urged to take a PCR coronavirus test but there are no plans for additional testing.
Meanwhile residents and workers across the whole Dudley borough are being asked to remain vigilant with rule changes next week set to allow people to meet people from outside their household indoors and hug friends and family.
Dudley Council said it would be monitoring the situation closely but that Public Health England's advice was not to set up additional testing.
Councillor Nicolas Barlow, cabinet member for health at the authority, said: "I would like to reassure people that at the moment there is no evidence to say that the variant puts people at greater risk but we do know it has a higher transmission rate, which is why it’s really important people get tested so that we can identify any further cases of this strain.
"This will help track undetected positive cases and prevent the spread of the virus any further.
"With restrictions easing next week, this is a reminder to us all that the virus is still with us and we need to proceed with caution. It’s really important people continue to follow hands, face, space and fresh air guidance, get tested regularly and have the vaccine when it’s offered."
A list of testing sites across the borough can be found on the council's website.