The majority of headteachers say reduced staff numbers, an inability to implement social distancing measures and a lack of personal protective equipment (PPE) could make it hard for schools to reopen to more pupils.
The government should improve its guidance on social distancing and provide the scientific evidence for pupils returning to class as "prerequisites" to schools reopening, heads say.
Coronavirus: The school opening options DfE is considering
Parents will not have the confidence to send their children back to class if schools are not given enough time to prepare, the leader of the NAHT school leaders' union has warned.
More than one in four – 29 per cent – of heads do not believe it will be possible to achieve some form of social distancing in primary schools, according to a survey shared with the PA news agency.
And fewer than one in 10 (9 per cent) think it will be achievable with children aged 7and under, the poll by NAHT suggests.
Coronavirus: The barriers to reopening schools
The survey, of more than 7,300 school leaders, comes amid speculation that schools could be asked to reopen their doors to more pupils before the summer holidays.
Prime minister Boris Johnson promised to deliver a "comprehensive plan" this week on how the UK lockdown may be eased and suggested that he would set out efforts to get children back to school.
On Sunday, Wales' first minister, Mark Drakeford, said schools in the country could be allowed to reopen their doors from the start of June in a phased approach.
In the survey, 82 per cent of school leaders said they could prepare for a reopening within three weeks, but 11 per cent said they would need longer.
One in four headteachers, questioned last week, said that fewer than 50 per cent of their staff were currently available to attend school as they were at risk, shielding or had caring responsibilities.
Schools, colleges and nurseries closed more than six weeks ago due to the coronavirus outbreak, remaining open only for vulnerable youngsters and the children of key workers.
Education secretary Gavin Williamson has said schools across England are set to reopen to children in a "phased manner" after the lockdown, but he has yet to set a date.
Asked what issues would prevent schools from reopening to more pupils by a date set by the government, the top answers among heads were an inability to implement appropriate social distancing measures (83 per cent), a lack of available staff (63 per cent) and a lack of PPE (57 per cent).
Marijke Miles, head of Baycroft School, a special school in Hampshire, told PA: "We are really worried about social distancing as a special school. It is very difficult to manage.
"We don't have any PPE in school at the moment, so that is a huge concern. There are aspects relating to the care of children and their behaviour which really require us to have effective PPE and a really good understanding of when and how to use it."
Asked what realistic actions they could take to achieve social distancing, school leaders said they were largely in favour of using rotas to limit the number of pupils (72 per cent), staggering lunchtimes and breaks (66 per cent), and rearranging furniture and seating in classrooms (60 per cent).
But some parents are still worried about sending their children back to school. Heads have been fielding calls in recent days from families who are concerned about their children's safety.
Simon Kidwell, head of Hartford Manor Primary School in Cheshire, has been contacted by families who are not ready to send children back.
He told PA: "If parents feel their children are safer at home, I certainly don't want to be enforcing school attendance with the usual levers that I have."
NAHT general secretary Paul Whiteman said: "If school leaders are not given sufficient time to prepare then we are going to have a very difficult return to schools.
"Parents and the public will not have confidence in schools' ability to keep children safe if they haven't been able to do the proper preparations to keep children apart and to make sure they have got all the right equipment there for children to wash their hands."
He added: "The endless speculation as to an exact date for schools reopening misses the point. It's not about when we return, it's about how we return.
"There are huge practical difficulties to solve, from a reduced workforce to the impossibility of social distancing at school to the lack of public confidence in the safety of their children returning."
A Department for Education spokesman said: "Schools will remain closed, except for children of critical workers and vulnerable children, until the scientific advice indicates it is the right time to reopen and the five tests set out by government to beat this virus have been met.
"We are also working closely with the sector as we consider how to reopen schools, nurseries and colleges, and will ensure everyone has sufficient notice to plan and prepare."