Telford & Wrekin's director of public health, Liz Noakes, is appealing to residents in Hadley, Leegomery, Oakengates and Ketley Bank to get tested as soon as possible.
The infection rate in the borough remains low, however in these areas the case numbers are higher compared to the rest.
A council spokesman said: "We want to do all we can to reduce the numbers further so please play your part."
It comes after cases of a mutation of the UK Covid-19 variant were identified among families in the Hadley and Leegomery area.
A test centre has been set up in the area today(24) and families in the area with school aged children are urged to take a PCR coronavirus test, even if they are not showing symptoms.
All who tested positive are now self-isolating.
Liz Noakes said: "Overall coronavirus cases in Telford and Wrekin remain low but we do need to closely monitor this mutation locally.
“Where we have identified these cluster of cases we are taking precautionary action to minimise and contain further infection. I would like to reassure parents and carers that all our schools are safe as Covid-19 safety measures are in place.
‘When cases are identified in schools this often indicates what is happening in the wider community – for this reason everyone in the local community needs to be alert.
“New variants and mutations of the Coronavirus underline why we all need to continue to play our part to help supress and control the spread of the virus."
Consultant in communicable disease for Public Health England West Midlands, Musarrat Afza said: “We are investigating a small cluster of cases of a variant strain of Covid-19 in the Hadley and Leegomery areas which are believed to be linked.
"PHE is monitoring the situation closely and all necessary public health interventions are being undertaken.
“For now, our advice remains the same: the best way to stop the spread of the virus is to observe social distancing and abide by the restrictions in place.”