Private school parents are demanding fee refunds, amid rising concern that some institutions could go bankrupt in the next few months.

Schools like Eton College, which charges £42,500-a-year and Uppingham School, which costs £38,720, will survive the current crisis due to long waiting lists.

Eton and Uppingham have both chosen to give parents a partial refund for this year’s fees.

Meanwhile, many other leading private schools have so far resisted pressure from parents to reduce or refund fees for this academic year.

Schools were told to shut their doors last Friday and send pupils home, with many now offering online classes and digital learning resources so pupils can keep up with their work from home.

Private schools are now billing for the summer term of fees, and there is a “massive clamour” among some parents for a fee refund, according to a source on the finance committee of one of the country’s most prestigious public schools.

“The reality is whether you are mega wealthy, middling or financially not well off, everybody in their different ways is exposed in this crisis,” they said.

“The many people who struggle to pay school fees will be under pressure. They are saying, ‘hang on, you can’t ask me to write a check for £12,000 or £13,000 if they are not even there’.”

The source told The Sunday Telegraph that a major financial strain on private schools is the fact that the overseas student market is “literally about to collapse”.

“The bottom line is, some independent schools will go bust between now and Christmas,” they said.

The Independent Schools’ Bursars Association (ISBA), which represents bursars at over 1,000 private schools in the UK and overseas, has written to their insurers demanding that they should pay out.

The insurers have told some private schools that “business interruption” insurance does not apply because Covid-19 is not one of their specified diseases.

But lawyers acting for the ISBA argue that they should be covered under “prevention of access” extensions when access to the schools is stopped by “any action by Government, Police or Local Authority due to an emergency which could endanger human life.”

The letter said that schools should be able to recover any loss of revenue “including as the result of schools having to refund or reduce fees”.

Stephen Spriggs, managing director at William Clarence Education which assists some of the UK’s leading public schools with overseas recruitment, said it is “absolute panic station” among registrars.

“I have spoken to 15 Registrars this last week alone, international applicants have fallen off a cliff,” he said.

“The regular channels of recruitment have stopped overnight: open days as well as exhibitions, events and trade shows abroad.

"People can’t be expected to commit to a £35,000-a-year fee, without seeing the school, and that is not realistically going to be possible for a long time.

“We are not talking about the Etons and the Harrows, they have huge waiting lists and they will be fine. But for mid-tier schools that are run on fairly fine lines anyway, any shock to the cash flow isn’t good.”

At the same time, British parents are “starting to push back” on the payment of summer term fees, he said, adding: “Schools are putting things online but £12,000 is hard to justify, there is a swell of opinion that they should be discounting."

Barnaby Lenon, chair of the Independent Schools Council and a former headmaster of Harrow School, has urged parents to resist the temptation to withdraw school fees, arguing that schools risk closure if they lose too much money.

David Woodgate, chief executive of the ISBA, said: “Schools in the sector are under immense financial pressure as a result of the COVID 19 crisis.

“They have, in very short order, put in place remote learning, homeworking for staff, and provided day care to children of key workers.

“Naturally, schools have a number of questions about insurance cover in this unprecedented time and these have been focused into one document as the start of a conversation with the insurance industry.”