School leaders should also think about opening profit-making franchises overseas to help meet the costs of running schools in the UK, members of the Independent Schools Council (ISC) were told.

Andrew Wynn, the bursar of Eton College, even raised the prospect of schools having to merge as the economic downturn hits independent education.

The warnings come as new figures released by the ISC show that its schools now collect more than pound;5 billion a year in fees, following a series of inflation-busting increases.

The 1,300 ISC schools, which teach more than 500,000 pupils, get almost 90 per cent of their annual pound;5.8 billion income from fees, the figures show.

Pru Jones, head of research at the ISC, said the numbers showed that schools were not “sitting on a heap of money and an ever-ready fund of riches”.

But heads at the ISC’s annual conference last week were told they should analyse ways of keeping fees under control.

Martin Weale, director of the National Institute of Economic and Social Research, suggested schools could even increase class sizes without parents knowing.

“Public schools feel very restrained about the extent to which they can let class size increase,” he said. “My guess is, you can squeeze 1 or 2 or 3 per cent more in there without people noticing too much.”

Nigel Richardson, head of the Perse School in Cambridge, said options for keeping fees down might be running out. Schools had already opened pre-preps and gone co-educational, and were now increasingly reliant on two parental incomes to meet fees, he said.

Mr Weale said: “The range of change you describe does make me wonder what the scope is for further development.

“Would you be happy about using a foreign operation to subsidise your core operation? It’s a question you ought to be asking.”

According to the ISC’s figures, its schools’ pound;5.8 billion turnover is more than is spent each year on electrical equipment, water supply and domestic appliances.