More councils in England are calling on the government to delay the reopening of primary schools amid rising cases of Covid-19.
Local authorities in Wolverhampton, Cumbria and Kent are now asking for a delay to the start of term on Monday.
Prime Minister Boris Johnson told the BBC parents should send their children to schools where they were open.
Some schools announced on Sunday they would remain closed after teachers said they felt it was unsafe to go in.
Mr Johnson told BBC One's Andrew Marr the risk to children was "very, very low" and the benefit of education was "so huge".
He added that while school closures would be kept "under constant review", the government would be "driven by public health considerations and by the massive importance of education".
Teaching unions have called for remote learning and some head teachers have begun legal action to force ministers to reveal data behind the decision for most schools to reopen.
All schools in London and parts of Essex, Kent, East Sussex, Buckinghamshire and Hertfordshire have been asked to stay closed until 18 January due to high rates of infection.
But Kent County Council leader Roger Gough has now written to Education Secretary Gavin Williamson, urging him to keep primaries in four districts closed in line with the rest of the county.
"Rates remain very high and in many cases were under strong upward pressure very recently. Kent as a whole now has a fairly even spread of high levels of infection," he said, urging that schools in Thanet, Canterbury, Dover and Folkestone and Hythe stay closed.
Ian Brookfield, leader of City of Wolverhampton Council, said he had "serious concerns" about schools opening safely.
"Unfortunately, the infection rate in Wolverhampton now stands at 530 people per 100,000 - the highest in the region. We are also seeing a very high positive test rate of 23% - similar to many areas of London and the South East," he said.
Cumbria's director of public health Colin Cox tweeted that he and and a colleague had also written to Mr Williamson, requesting that the county's primary schools be kept closed.
He said infection rates in some parts of the county were "doubling every 4-5 days".
Following extensive discussions over the last 48 hours, the CCC Exec Director (People) and I have this morning jointly written to DfE formally requesting that Cumbrian primary schools are added to the Contingency Framework of schools not expected to open tomorrow.— Colin Cox (@CumbriaDPH) January 3, 2021
Essex County Council has written to all primary schools in the Colchester, Tendring and Uttlesford districts advising them not to fully open on Monday or Tuesday and to move to remote learning.
This would bring them into line with the rest of the county, where schools will remain closed.
Leaders in Birmingham, Manchester, Lancashire, Slough and Gateshead said on Sunday they would support head teachers who did not think it safe to open.
A Norfolk County Council memo to schools seen by the BBC acknowledges that head teachers "may find it difficult to be certain that you will have sufficient staff to open safely on Monday".
It urges them to make a decision on whether they can reopen "as early as possible so that you can communicate with parents and carers".
'Teachers are key workers'
Southampton City Council leader Christopher Hammond said he had advised primaries to prioritise in-school learning for vulnerable children and the children of key workers if they did not have enough staff to operate fully.
James McInnes, cabinet member for children's services and schools at Devon County Council, said teachers and staff should be prioritised for vaccinations against Covid-19.
"Teachers are key workers and I really think they should be right at the front of the queue in terms of getting a vaccination because they're keeping children in school," he said.
Forty members of staff at All Saints Church of England Primary School in Bradford contracted coronavirus last term, head teacher John Davie said. He also thinks vaccines are paramount.
"The government needs to make it safe for the staff," he said. "They've doing that in care homes and hospitals by providing vaccinations, they should do the same for us in schools so we're safe as well.
"Last term, 40 members of staff contracted Covid, if they'd been working at home I don't think that would have been the case."
On Saturday, Brighton and Hove City Council advised primary schools to switch to remote learning next week.