The youngest children are returning to the classroom as schools reopen to more pupils.
Children in primaries one to three are due back in Scottish schools from Monday, along with some senior secondary pupils who need to do practical work for qualifications.
All children under school age in early learning and childcare are also returning.
Senior secondary pupils will need to stick to two-metre social distancing within schools and on school buses, while Covid-19 testing will be made available to them and teachers.
He acknowledged that the Government has little “headroom” regarding the R number – the rate at which the virus grows.
Mr Swinney told the BBC’s Sunday Show programme: “We’ve all got to play a part, not just the school environment but the whole surrounding environment of society, to make sure that we do all that we can to suppress the virus.
“Because we don’t actually have an awful lot of headroom between where the R level is just now – somewhere between 0.7 and 0.9 – and 1, where obviously the virus begins to spread exponentially.”
The Education Secretary said on Monday the biggest threat to education would be an increase in the spread of the virus across the country, as he urged Scots to follow public health advice.
He told Good Morning Scotland: “The biggest threat to the opening of schools is not outbreaks in schools, it is community transmission of the virus.”
When asked if parents at school gates contributed to prevalence of the virus before this lockdown, Mr Swinney said: “The whole community was driving the virus.
“I don’t particularly want to single out particular groupings – the whole society was interacting too much, that’s why we had to go into lockdown.”
Mr Swinney said ministers would be monitoring data carefully when the initial cohort of pupils returns before deciding on whether others could go back to class.
Professor Devi Sridhar, chairwoman of global public health at the University of Edinburgh, told BBC Good Morning Scotland on Monday: “I think we should keep perspective.
“There will likely be cases emerging in schools over in the next few weeks but the vast majority of schools should be fine and that’s what we have to keep perspective on.”
Prof Sridhar, who is an adviser to the Scottish Government, said: “If you look at the vast majority of scientific knowledge on this, from the United States from across Europe, from East Asia, the view is really that children are not driving transmission.
“They can get infected definitely and they can infect others but they are much less likely to and so if we talk to paediatric colleagues about this they are not finding a lot of asymptomatic infections in children and they are testing quite regularly in hospitals.”