The Shropshire, Telford & Wrekin Sustainability and Transformation Partnership (STP) said it would put in place measures to avoid a local lockdown if there was a significant increase in cases.
The partnership was responding to reports that people had been sent to the site near Ironbridge from hundreds of miles across the country last week leading to traffic jams near the site off Jiggers Bank. Staff ran out of test kits and turned away people arriving for booked tests.
It said the public should only book tests if they are displaying symptoms of Covid-19 such as high temperature, a new continuous cough or a loss or change to sense of smell or taste as most infected people with have at least one of these symptoms.
If those with symptoms get a test this will help to stop the spread of the disease in the community.
Those without signs should not book tests as they could be taking a test slot away from someone who really needs it as autumn and winter approach and more people will get colds and the flu.
The service said there was a high demand for testing across the UK and that it was seeing more people trying to ring 111 about testing, but it needed to ensure that 111 is protected for people who are ringing about other medical and health issues.
A statement said: “Across Shropshire, Telford & Wrekin we’re continually updating our local outbreak plans and will put in place measures to avoid a local lockdown should we see a significant increase in cases locally.”
Tests should be booked in advance online or by ringing 119.
Anyone without an appointment who turns up at a testing site will be advised to book an appointment.
Residents are urged to keep trying their local centre for slots.
Health Secretary Matt Hancock has pledged to find what caused the problems.