Unite, which represents over 80,000 bus drivers in the UK, said the measures should start in London, where 27 bus workers have died from the disease.
The union said all bus workers should have their temperature checked as they arrive for work, with anyone registering a high temperature given help to access a test, which they are entitled to receive as key workers.
London regional secretary Pete Kavanagh said: "By checking temperatures and ensuring drivers are then tested, not only will we be able to reduce the risk of infection but we can also ensure that workers can quickly return to work if the test is negative.
"Once it is identified that a driver has a high temperature the company has a duty of care. The driver is instructed to go home and get tested and further contamination is avoided.
"If it protects just a handful of workers from an infection that has killed far too many bus workers in London then it is well worth doing."
But Confederation of Passenger Transport UK, which represents bus operators, said staff had access to tests as key workers.
Scotland director Paul White said: “Keeping bus workers safe is a number one priority for operators.
“In line with government guidance, operators are encouraging employees who display Covid-19 symptoms, or who live in a household where someone has symptoms, to self-isolate and take a Covid-19 test.
“Bus workers already have access to the dedicated Covid-19 testing arrangements for key workers.
“Operators are also ready to use the employer referral system to help bus workers access online tests where they cannot book one themselves.
“We shall continue to monitor and take guidance from the government on this matter."
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