Scientific advisers to the government told ministers they should have an effective track, trace and isolate strategy as they reopen schools.
Prime Minister Boris Johnson wants schools to reopen on 1 June and there is doubt over whether such a system will be in place.
The advisers have also said that teachers do not appear to be at a greater risk of catching COVID-19 than other professions – but there is still some risk if schools reopen.
The advisers also suggested some form of testing in school takes place.
They also advised that it is safe to reopen schools as long as the speed of infection, known as the R rate, is below 1 – a condition which is currently being met.
News of their advice comes as SAGE released its reasoning for how schools could be reopened safely.
Downing Street has faced growing calls to release the science behind its move to potentially begin a phased reopening of primary schools from 1 June.
Under plans to ease the coronavirus lockdown unveiled by the PM earlier this month, children in nursery, reception, Year 1 and Year 6 will return to their classrooms.
A final decision on whether to go ahead with reopening schools is expected to be taken by the government on or before 28 May, after the most up-to-date scientific evidence has been reviewed.
Mr Johnson has promised a “world-beating” track and trace system to stop a second COVID-19 peak and help ease the lockdown will be in place by the time schools begin to reopen.
But the push to reopen schools has sparked opposition from some quarters, with more than two dozen councils saying they will not begin allowing pupils back from that date.
Unions have also expressed concerns about whether teachers, support staff and pupils will be safe.
Although a number of councils have expressed opposition to the potential reopening of schools, many are not under local authority control.
Academies, state schools that are not run by councils, could take their own decisions on whether to follow the government’s plans.
A number of chief executives of academy chains have said schools have to reopen soon in order to avoid “irreparable” damage to vulnerable children.
The chief executive of REAch2, the largest primary academy chain in England, has said it plans to reopen all its schools for the priority year groups identified by the PM.
The devolved nations have taken a different approach to schools than Westminster.
In Scotland, the plan is to have all children back on a part-time basis on 11 August.
In Northern Ireland, the aim is to restart classes for all children in September if enough progress has been made in tackling COVID-19.
Meanwhile, in Wales, the government has not given any dates for when the country can expect schools to reopen.