South Lakes MP Tim Farron and local Liberal Democrat councillors Nick Cotton, Suzie Pye and Ian Mitchell have asked for COVID-19 testing to be offered to everyone in Sedbergh after an outbreak at Sedbergh School.
At the weekend, Sedbergh School reported that a 'significant' number of their pupils have received a positive test.
The whole school is being tested this week via a Mobile Testing Unit led by the UK armed forces.
Local district councillor Suzie Pye asked that the Mobile Testing Unit is made available to the whole town.
Suzie said: "This is a worrying development and it's really important that testing of staff and pupils is prioritised.
"I want to pay tribute to how the school has responded to this situation and the measures they've taken to help isolate any pupil that receives a positive test and therefore helping to protect our community.
"Sedbergh School is a massive part of the town and many people employed there live in our community, so there is a very strong case in offering a test to any local resident who would like one.
"We must act quickly to protect local people and get this under control."
Cumbria County Council say they have considered the request for mass testing of the town, but at this point in time, they will not take the request forward.
Colin Cox, Cumbria’s Director of Public Health, said: “We considered this request carefully but based on the evidence we currently have, mass testing of the whole local population is not warranted. We are continuing to work closely with the school to support their management of the outbreak and the situation remains under review.