More than 2,000 English primary schools could stay closed on June 1 despite government reopening plans - with dozens of councils not confident about safety.
The government has targeted the start of next month to begin bringing youngsters back into the classroom.
But it can only happen if scientists say it is safe to do so, ministers say, and many education bosses are yet to be convinced.
Liverpool, Hartlepool, Birmingham, and even Tory-run Solihull and Essex are among those warning June 1 might not be possible.
Their stance means a worst-case scenario of more than 2,000 schools not being ready to reopen in 10 days' time.
The government's June 1 opening date coincides with the point at which Boris Johnson has promised that the UK's 'track-and-trace' system will be in place and able to track 10,000 cases a day.
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The government plans to bring reception and Years 1 and 6 pupils back to class, with other primary years returning later next month.
Some local authorities are looking at later dates, while some are still waiting to see what will happen.
Others are suggesting returns may be part-time - despite the government saying this isn't an option.
The number of councils not expecting all their primary schools to be ready to reopen by June 1, and therefore set to rebel range from 25 to more than 35.
Estimates for the number of schools set to stay shut varies, with The Sun saying up to 2,200 could remain closed.
Some councils, including Sunderland and Rochdale, have suggested they may not follow the Government's time frame while other local authorities have already ruled out reopening on June 1.
Richard Watts, leader of Islington Council in north London, said on Wednesday that the council did not support rushing to hit an "arbitrary deadline" set by ministers, adding that he recognised the "grave concerns" from parents.
Other local authorities say a phased approach will be needed next month to minimise safety risks.
Sandwell Council in the West Midlands has said not all children in nursery, reception, year one and year six will be able to start "on a full-time basis" from June 1 due to the size of classrooms.
The findings come as the Local Government Association (LGA), which represents councils in England, called for access to coronavirus testing data in order to safely reopen schools in different areas.
Mark Woolhouse, professor of Infectious Disease Epidemiology, University of Edinburgh, told the PA news agency there are three main risks associated with children returning to school.
He described these as the risk to children, the risk to teachers, and the risk of transmission in the community increasing.
Speaking as an independent researcher, and not in an advisory role, he said: "Covid-19, though a very unpleasant virus, and capable of causing illness, on occasion in any age group, in fact, is very, very, very rarely a serious problem in children.
"Staff are of course adults so they are more vulnerable to infection, more vulnerable to symptomatic infection."
But he added that how vulnerable teachers are in school depends on whether they are considered a major source of outbreaks.
"And so far, the indications from around the world are they are not," said Prof Woolhouse.
Prof Woolhouse said he knows of no evidence that suggests school staff are at greater risk in the workplace that any other adult in a workplace.
