Schools "will not be able to cope" with the number of pupils who could potentially arrive on Monday morning, headteachers have warned.
Parents included in the government's key workers list should only send their children to school "if there is no safe alternative" otherwise schools could be "overwhelmed", according to the Association of School and College Leaders (ASCL).
The coronavirus pandemic has led to schools being closed after today to all but the children of key workers and vulnerable pupils.
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A list identifying these key workers was published by the government just before midnight last night.
It includes doctors, nurses, care workers, teachers, people responsible for the management of the deceased and journalists working for public service broadcasting.
Coronavirus school closures
But Geoff Barton, ASCL general secretary, said if everybody who is "potentially covered by the list" sent their children to school next week, the numbers would be "significant", and schools would be "overwhelmed".
He said: "The key worker list is extensive, and schools will be operating on very reduced staffing.
"We already know from the trajectory of this that we're not going to have many staff.
"If everybody who is potentially covered by the list sent their children to school on Monday the numbers would be significant, and schools would be overwhelmed, particularly small schools.
"We are therefore advising that these parents should only send their children to school if there is no safe alternative.
"We are advising our members that it is their decision which parents they offer this option [sending their child to school] to.
"If they receive additional requests then it is their call on whether or not to accept that request. We will support their decision."
He added that, while "schools will endeavor to do their best to provide continuity of learning for all children", provision on-site is "likely to be more akin to childcare than a normal timetable".
"It is important that the public understands that this is not business as usual," Mr Barton said.
"Schools are working to an incredibly tight timescale to turn around this provision and we would ask everybody to show patience and understanding in this extremely challenging situation."
Meanwhile, teaching union the NASUWT is calling for teachers to be given priority when Covid-19 tests become available.
Chris Keates, acting general secretary, said: "Giving priority to teachers to be tested for Covid-19 and access to personal protective equipment would help to reassure many that keeping schools open for priority groups of pupils is not only necessary but also the responsible thing to do."