Telford & Wrekin Council said that the infection rate for the borough is at one of its lowest levels for months – with 14 per 100,000 people.
It comes after the confirmation of cases involving an Indian variant of Covid in some pupils attending Burton Borough School in Newport.
A testing programme has been launched in the area, with people also being encouraged to take up the option of a vaccination if offered.
Councillor Andy Burford, Telford & Wrekin Council's Cabinet Member for Health and Social Care, said: "News that we have identified one of the Indian variants in our borough will cause some concern.
"However, the testing we are taking is precautionary to prevent any further spread.
"After the first day, we have been pleased with the number of people coming forward to get tested.
"I'd like to say a big thank you to everyone who has played their part and got a test when asked.
"Only through regular testing and isolation of positive cases can we help to keep others safe."
The council said that in total there have only been 25 new Covid cases reported for Telford and Wrekin in the seven days up to May 16.
It is nine more than the previous seven-day period but the borough's infection rate is lower than the national average.
Liz Noakes, Telford and Wrekin's Director for Public Health, said: "Our cases are some of the lowest we have seen for a while.
"However, we are working to respond quickly to outbreaks, including one involving one of the Indian variants.
"Through testing and the isolation of positive cases, we are keeping cases low and the virus under control.
"Thank you to everyone who has got a test when asked. Each one helps us to track and stop the spread of the virus."
As part of the response to the variant cases the council has asked all secondary-school aged children in Newport, staff at the schools and their extended families and support bubbles – from Burton Borough, Haberdashers' Adams Grammar and Newport Girls' High School Academy – to take a PCR Covid-19 test.
Families with primary-school-aged children in Newport and all other Newport residents have also been encouraged to get PCR tested as soon as they can.
A testing centre is open in Newport at Shuker Playing Fields, Sports and Social Club, Barnmeadow Road, TF10 7TP.
It is open from 11am to 6pm, Monday to Friday, and 9am to 3pm over the weekends, with no appointment needed.