Dr Patrick Roach, general secretary of the NASUWT teaching union, has warned the government must prioritise the education sector for the allocation of tests.
In a letter to the schools minister, Dr Roach said the union had heard of approximately 600 pupils being told to self-isolate in one local authority area and he said the “number is growing”.
There have been widespread complaints that the UK's testing regime is inadequate, with some people being forced to drive for hours to get a test, and others having to wait days for appointments or returned results.
The union leader told Nick Gibb, Minister for School Standards, that pupils who have been sent home with symptoms are “facing uncertainty” about when or where they will be able to access a test.
He warned that the delays in testing have meant some students and staff who are part of a “bubble” within a school are not being isolated even where there are multiple suspected cases.
“This is putting at risk the health and safety of others within the school and within the local community,” Dr Roach said.
Schools have been hit with Covid-19 cases since it became compulsory for pupils to return in England and Wales from earlier this month.
Some have been forced to close their doors days, while others have told whole year groups and classes to self-isolate for two weeks following confirmed cases.
NASUWT members are “expressing serious concerns about the failure” of the testing system and the impact on schools, the letter to Mr Gibb says.
Dr Roach said, in particular, areas hit with local coronavirus restrictions have been "unable to cope" with testing demands.
“Teachers, support staff and children and young people are unable to access tests where they have Covid-19 symptoms.
"Employers are struggling to deal with the implications and consequences," he added.
Local authorities across the country – including in the North West of England – are struggling to cope with the demand for tests from pupils and school staff, the union suggests.
The letter identified Bury, where 600 pupils are self-isolating, and Salford, where the council has been inundated with requests for tests from schools.
Dr Roach said: “Schools appear to be seeking to do their utmost to carry on.
“However, we have reports that schools are unable to cope with a situation that is becoming increasingly out of control.”
The founder of Oasis Community Learning, responsible for 31,500 children at 52 academies across England, said 1,200 pupils had been sent home over the first six days of the new school year.
Writing in The Sun, Steve Chalke added: “The reason is either pupils or teachers have symptoms and can’t return until they get a negative test result.”
Earlier this week, Dr Roach warned Education Secretary Gavin Williamson of the possibility of legal action if the government fails to protect teachers working in schools.
In a letter to Mr Williamson, Dr Roach demanded that stronger protections are put in place in schools which opened their doors to all pupils full-time this month.
The Education Secretary is set to face questions from MPs on the impact of the Covid-19 pandemic on pupils - including the grading chaos in England over the summer.
The health secretary has previously it could take "weeks" to resolve issues around testing and has admitted there are "operational challenges" related to a surge in demand.