The government and unions should "stop squabbling and agree a plan" to reopen schools safely, the Children's Commissioner for England has said.
Anne Longfield said many disadvantaged children were losing out because of schools being closed for so long.
Teachers' leaders met the government's scientific advisers on Friday, but did not agree how to return to the classroom safely.
Some local authorities have said they will not reopen schools from 1 June.
The government has set out plans to begin a phased reopening of primary schools in England from 1 June.
But the plans have been challenged by teachers' unions, who have disagreed with the Department for Education over whether it is safe to return to school.
Kevin Courtney, co-leader of the UK's biggest teaching union, the National Education Union (NEU), told BBC Breakfast the meeting with scientific advisers on Friday was a "step forward" but that more cooperation was needed between the government and unions to "find a safe way back".
The NEU has drawn up five tests which, it says, the government should meet before schools reopen. These include regular testing, protection for vulnerable staff and a national plan for social distancing.
Meanwhile, Liverpool's schools will not reopen until mid-June "at the very least", the city's council said.
Steve Reddy, director of children and young people's services in Liverpool, said: "Our guiding principle is that schools can only reopen to other pupils when it is safe to do so and not a moment before."
Hartlepool Borough Council has also challenged the government's timetable, saying schools will not reopen at the start of next month "given that coronavirus cases locally continue to rise".
However, the mayor of Middlesbrough said that, although children and teachers have to be kept safe, "no-one can ever be given a 100% guarantee", stressing the impact that school closures were having on deprived children.
"It's all about reasonable risk," Andy Preston told BBC Breakfast. "And my goal is that we are in a situation where I think people are reasonably safe, I think that I can hand on heart say that I would send my kids into that school, that I am willing to go into that school."
'It will cause children damage'
Teachers have said they are worried about the emotional distress returning to the classroom could have on both staff and pupils.
Becky, who has asked her surname not be used, teaches Reception and Year 1 at a primary school in Birmingham.
Speaking about the effect new rules will have on children, she said: "They won't be able to play with toys, play with their friends. When they're distressed and upset, how can we comfort them from a distance? It will cause them damage."
Oliver Jarman, a secondary school teacher in Yorkshire and a father of four, said: "We need more clarity about the science behind the decisions the government is making. I'm saying that as a teacher and a parent."
While many primary schools in England are under local authority control, others are run by academy trusts.
Steve Chalke, founder of the Oasis Trust, which has 35 primary schools across the country, said opposition to reopening fails to recognise the harm to disadvantaged children from missing school.
England is the only UK nation to set a date for schools to start to return.
The British Medical Association has backed teachers' unions by saying Covid-19 infection rates are too high for England's schools to reopen.
The infection rate in the UK - the so-called "R-number" - has crept up from between 0.5 and 0.9 to between 0.7 and 1.0.
The number needs to be kept below one in order to stay in control and any increase in the number limits the ability of politicians to lift lockdown measures.
Prof Sir Mark Walport, the government's former government chief scientific adviser, said the rise in the R-number - albeit "relatively small" - is concerning and illustrates how carefully social distancing measures need to be relaxed.
"There's no question that the prospect of a second wave still exists," he told BBC Radio 4's Today programme.
Speaking about schools reopening, Prof Walport said schools have returned "in a very cautious fashion" in Denmark for the past month and the country's R-number has reduced.
The BMA has said opening schools should not be considered "until we have got case numbers much lower".
But Ms Longfield said schools needed to open "as quickly as possible" and called for stronger safety measures to be introduced, such as regular testing for pupils and teachers, managed interaction between adults and deep cleaning of facilities.
"I don't think we are ever going to get to the stage without a vaccine that this is going to be 100% safe," she told BBC Breakfast.
"I am disappointed that the debate about when some primary school kids can return has descended into a squabble between government and the teaching unions," said Ms Longfield earlier.
"All sides need to show a greater will to work together in the interests of children."
Schools have been closed by the coronavirus for most pupils since 20 March, staying open only for the children of key workers and vulnerable children.
But even many children in these categories are not currently going to school - with about 2% of pupils currently attending.
Ms Longfield said that deprived and vulnerable children would suffer most from missing school and there was a growing "disadvantage gap".
"We know that the longer schools are closed the greater the impact will be on social mobility and that many children are really struggling without seeing their friends and the structure that school brings.
"We need to face the reality that for a number of reasons there are hundreds of thousands of children who can't access meaningful education at home."
She backed plans to bring back children in Reception, Year 1 and Year 6.
"But we should have an aspiration that all children return to school in some form before the summer and that school buildings are used for activities, summer schools and family support over the holidays."
A Department for Education spokeswoman said: "Getting children back to school and nurseries is in their best interests and all those working in education have a duty to work together to do so.
"We welcome the children commissioner's support for a phased return of children to primary school with many of the measures she raises, like staggering drop-off and pick-up times, keeping children in smaller groups and regular hand washing, already in train."
Schools in Wales will not be going back on 1 June and it is not expected that schools in Scotland or Northern Ireland will go back before the summer break.
Source: BBC News