Support for a fixed date for reopening schools to more pupils is "vanishing quickly", heads have said.
The government must show "greater flexibility" when it comes to planning for schools to reopen more widely, and should allow decisions to be made on a local basis, according to school leaders.
Paul Whiteman, general secretary of school leaders' union NAHT, made the comments shortly after the government published the scientific evidence underpinning its decision to reopen schools to more pupils from 1 June.
Independent SAGE: June 1 'too soon' to open schools
The documents were released today by the government's Scientific Advisory Group for Emergencies (Sage).
Mr Whiteman said: "Support for a fixed date for school return is vanishing quickly. What is needed now is local flexibility to determine when it is right for schools to open up to more pupils, informed by evidence of what is happening in their own local area.
"We have never expected certainty, all we have asked for is clarity. The publication of the evidence being used to inform the government's decision making is an important step in achieving greater transparency.
"The government needs to show greater flexibility and a willingness to take local circumstances into account."
Mr Whiteman added that, having seen the evidence from Sage, NAHT is calling on the government to explain why it has "so strongly asserted" that returning to school on 1 June is a "wise thing to do".
Geoff Barton, general secretary of the Association of School and College Leaders, urged all those involved in the debate about when schools should open "to desist from cherry picking pieces of scientific evidence from lengthy and complex documents to bolster their own agenda".
He added: "We are pleased the government has published this scientific evidence. It will quickly become clear to anybody reading the papers that the science is not definitive, and indeed, it cannot be in a situation where we are dealing with a new virus which is not fully understood.
"So, this is about making the best judgements on the basis of what is known, and exercising the utmost caution about how we proceed.
"This means the government must be able to show very clearly that its five tests have been met before it gives the green light to any wider opening from 1 June, and it must then be flexible about the timescale and approaches needed to bring in eligible pupils."
It was also revealed today that scientific advisers to the government believe that an effective test, track and trace system must be in place before schools reopen to more pupils.
Sage has advised the government that, in order to enter the next phase of the lockdown plan – in which schools would be partially opened – a test, track and isolate system should be established.
Mr Whiteman said: "A robust test, trace and isolate policy is essential if we are to successfully return more pupils to school."
Patrick Roach, general secretary of the NASUWT teachers' union, added: "The evidence presented by Sage in terms of justifying the decision by government to start to reopen schools from 1 June is inconclusive.
"The papers highlight the significant gaps in evidence, knowledge and understanding which remain in terms of the susceptibility of children to Covid-19 and how infectious those with mild and asymptomatic cases of the virus may be.
"The committee states that large-scale community testing is needed to better understand and monitor the prevalence of and susceptibility to Covid-19 in children, yet the government’s plans for the reopening of schools from 1 June are premature whilst a widespread community testing system will not be in place.
"The Sage papers published today will only add to teachers’ uncertainty and anxiety.
"Importantly, the committee has concluded that interventions around preventing the spread of the virus must be eased in a logical manner.
"However, the arrangements for easing the current restrictions on schools have been far from logical and are yet to secure the confidence of parents and school staff.
"The NASUWT remains of the view that no school should reopen until it can be demonstrated that it is safe to do so. We remain ready to work with the government on a way forward which will ensure that staff and pupils can return to schools safely."
A Department for Education spokesperson said: "The evidence published today shows we looked at a range of options in drawing up this phased approach to reopening schools, which is in line with the government’s overall 'roadmap'.
"As in other countries across Europe, the first phases of the wider opening of schools will prioritise younger children.
"Advice from Sage shows there is a lower overall risk from opening schools and nurseries to younger children, and that they are less likely to become unwell if infected with coronavirus compared to adults.
"This cautious, phased approach for allowing a limited number of pupils back into classrooms has been, and will continue to be, informed by the best possible scientific and medical advice."