It comes as a public health official warned parts of the county are seeing the start of a "third wave" of Covid.
Telford & Wrekin Council and the Red Lion pub in Madeley are working together to monitor the situation after being informed that those who recently tested positive had visited the pub.
In response, the council is launching a precautionary testing drive which is asking pub staff and visitors who went to the pub between May 23 to 31 to get a test.
People should get a test even if they have been vaccinated and have no symptoms.
Self-isolation is only required if people have the three main symptoms – a new continuous cough, high temperature, loss of smell or taste.
Liz Noakes, Telford and Wrekin's director for public health, said: "We are launching a precautionary testing drive to help identify cases and prevent further spread.
"Getting a test helps to keep cases low and everyone safe.
"We know similar testing drives in Newport and Hadley before have helped to prevent further spread.
"So, please get your precautionary test as soon as possible - we must all play our part."
A spokesperson at Marston’s, which operates the Red Lion, added: "We are working closely with the retailer of the Red Lion and the local authority environmental health officer following a small number of positive cases of coronavirus linked to customers who had visited the pub.
"The pub has been advised to remain open following a deep clean of the premises and PCR testing for all staff members."
People can book a test through the national booking system.
When booking a test, select the option 'My local council or health protection team has asked me (or someone I live with) to get a test, even though I do not have symptoms'.