Some schools are struggling to find any supply teachers to cover lessons amid staff shortages, while headteachers are having to cope with more families withdrawing their children.
Geoff Barton, general secretary of the Association of School and College Leaders, warned schools have been forced to partially close as it has become “increasingly difficult” to stay open.
The largest teaching union in Europe has also called for school closures “at least for some time and at least in some areas” as it said the government has failed to justify its current policy.
A lack of government advice for schools is creating “chaos and confusion” during the coronavirus outbreak, another teaching union has said.
In one school in Bristol, a fifth of its staff stayed at home and there were not any supply teachers available locally.
“Simply expecting all schools to remain open is becoming increasingly untenable and will end up being counterproductive because closures will happen without coordination,” Mr Barton said.
The school leaders’ union is hearing that up to 20 per cent of teachers in some schools are already self-isolating and the number is “rapidly rising” following the prime minister’s advice on Monday for everyone in the UK to avoid “non-essential” travel and contact with others to curb coronavirus.
Official advice from the government says schools should remain open – but a number of schools across the country announced plans to close their doors on Tuesday because not enough staff could go to work.
The government has urged pregnant women and people with certain health conditions to self-isolate and stay at home as part of a range of measures to try to control the spread of the virus.
The National Education Union (NEU) said there was an “apparent contradiction” between banning mass gatherings and keeping schools open in a letter to Boris Johnson on Tuesday.
Mary Bousted and Kevin Courtney, NEU joint general secretaries, said: “Given your failure to release modelling comparing different scenarios of school closures, we are now forced to call on you to close schools, at least for some time and at least in some areas.”
They added: “We intend to advise all our members in [vulnerable] categories or caring for people in them to stop attending schools and colleges from next Monday at the latest. Some will do so earlier.
“This will make the running of schools all the more complicated.”
The union has also called for Sats exams to be abandoned and for the government to release proposals on the “inevitable widespread disruption” to GCSE and A-level exams.
The NASUWT union, which also represents teachers, said a lack of advice was creating “chaos and confusion” and placing “intolerable pressure” on staff.
Acting general secretary Chris Keates said: “All of the announcements continue to be couched as guidance or advice, which is simply serving to increase anxiety and uncertainty.”
Schools are struggling with diminishing staff levels, while changes to staff working conditions have the potential to compromise health and safety for staff and pupils, she said.
Ms Keates urged for a definitive decision from the government, adding: “This situation cannot be allowed to continue.”
But education secretary Gavin Williamson has insisted that the current medical and scientific guidance says that schools and other educational settings should remain open.
In a video posted on Twitter, Mr Williamson said: “The chief medical officer has said the impact of closing schools on children’s education will be substantial, but the benefit to public health would not be.”
He added: “I have been in discussions with unions and school leaders to minimise disruption for schools and help them to stay open to support our young people.
“I know that staff shortages are presenting teachers and headteachers with great challenges and I want to say how deeply grateful I am for their civic and community leadership.”