Almost half the respondents said that greater collaboration between private and maintained schools would improve state education. And 54 per cent of state school parents said private schools should do more to share their expertise.

The findings come as the Charity Commission prepares to publish its final guidance on public benefit tests this month. It is expected that schools will have to offer more places to children from poorer families and carry out annual audits of how they are contributing to the public benefit in order to keep tax breaks worth pound;100 million a year.

After publishing its general guidance, the Charity Commission will begin consultations with independent schools on specific rules.

The survey of public attitudes to independent education, carried out by Zurich Insurance, revealed that only 10 per cent of people disagree with moves to make private schools more accountable.

Over a third of respondents said giving bursaries to pupils from poorer families was the best way to meet public benefit tests. Nearly a fifth said opening up school facilities to the community was the key.

Among parents of private school pupils, 40 per cent believed such schools should increase means-tested places. But 69 per cent did not want their fees to fund this.