GMB is calling for ministers to bring forward a testing regime for every school in Scotland.
A survey from the union found more than two-thirds of support staff in schools and nurseries do not feel safe in the workplace.
The union has written to Deputy First Minister and Education Secretary John Swinney urging him to follow the interventions made in the social care sector by giving workers the opportunity to access testing in their workplace.
The survey of more than 1,400 members - who are employed in roles such as cleaners, janitors, caterers and pupil support assistants - found 96% believed regular testing should be offered at work.
Almost two-thirds (63%) said they felt unsafe while more than a fifth (23%) have confirmed a suspected or positive case of Covid-19 at work.
GMB Scotland organiser Helen Meldrum said: "Over a week since the return of Scotland's schools, it's clear that many support staff do not feel safe at their work and an overwhelming majority of them want to be able to access a Covid test at work.
"If support staff need to book a test, they must absorb the financial costs and time implications to do so, and for a chronically low-paid workforce with many employed on multiple contracts across multiple workplaces, that's just not credible.
"The failure over the summer months to listen to the voices of school support staff echoes what we witnessed in care earlier this year, and in this case the political focus has been on teachers and pupils while support staff have been forgotten.
"You cannot have a safe return to full-time education if a significant chunk of the workforce needed do not feel sufficiently safe, valued or heard by our decision-makers, and that's why we urged the Deputy First Minister to intervene now."
It comes as more cases of coronavirus were confirmed in the cluster surrounding Kingspark School in Dundee on Monday.
NHS Tayside said the number of positive cases had reached 27, up from 22 on Sunday. Of these, 21 are members of staff, two are pupils and four are community contacts.
The school has been closed since Wednesday evening.
A Scottish Government spokeswoman said: "Guidance, informed by the latest scientific advice, sets out the range of measures schools should take to minimise the risk of the spread of the virus.
"We know concerns still exist and we want teachers and staff to not just be safe, but to feel safe, in school and in nursery.
"Teachers, nursery and school staff can already access a test for Covid-19 on demand through their employer if they are concerned they have been at risk from infection.
"We have written to every local authority providing information on how staff can access a test."