Health chiefs at Shrewsbury and Telford Hospital NHS Trust, which also runs Royal Shrewsbury Hospital, say staff and patients are being tested as a precaution and contact tracing is being completed when necessary.
Hayley Flavell, director of nursing, said the trust is also working closely with neighbouring hospitals at Stoke and Wolverhampton, and with the ambulance service for the transfer of appropriate patients.
The measure is being reviewed on a daily basis.
Hayley said: "A small number of patients at PRH have tested positive for Covid-19.
“Where necessary, staff and patients have been tested as a precaution and contact tracing is being completed.
“Appropriate infection prevention and control measures are in place, and we are working with our local authority public health colleagues, Public Health England and our partners across the health system."
Medical director, Dr Arne Rose said the trust's hospitals are not yet seeing the same rise in coronavirus cases as some elsewhere in the country.
'Small but increasing number'
But he added that they were treating "a small but increasing number" of patients who have tested positive for Covid-19.
He is encouraging people to take precautions and keep following the Government's guidance in a bid to stop the spread of the virus.
“We know that, in England, Covid-19 cases are increasing in every age group and that is why it is vital that people continue to follow the latest guidance to reduce the spread of this disease," he said.
“Please, continue to wash your hands regularly and thoroughly; wear a face covering when you are in an enclosed space and maintain social distancing.
"This is an important phase in our fight against coronavirus, and it is up to all of us to play our part.”