Kathy August is expected to use her speech at the Headmasters’ and Headmistresses’ Conference in St Andrews to call for better co-operation between the two sectors.

Mrs August became principal of the Manchester Academy last year, one of three new academies sponsored by the United Learning Trust, an Anglican charity.

The charity also sponsors academies, semi-independent state schools, in the London borough of Lambeth and Northampton and is planning more in London, Salford, Barnsley and two in Sheffield.

The trust was created by the Church Schools Company, which already runs eight private schools. Mrs August said staff at the trust’s academies had exchanged ideas with heads of the eight Church private schools.

The Government is courting private schools as potential sponsors of some of its planned 200 new academies. King’s school in Canterbury, Oundle in Peterborough and Dulwich college in south London have all expressed an interest in sponsoring academies.

But critics claim academies are selective and damage neighbouring comprehensives. Last month unions at the TUC conference denounced the scheme.

But Mrs August said: “Our academies, certainly, do not exercise any selection and I think if the independent sector can help us get the best deal for our youngsters, then we should be embracing that chance. No parent or child has ever come to me saying ‘We like being here but we are worried because you are friendly with the independent schools’.”

Dr Martin Stephen, chairman of the HMC, is likely to reinforce the calls by demanding more state support to enable bright children to study at top private schools. Labour axed the assisted places scheme - bursaries for pupils from low-income families - when it came to power in 1997.

But Liz Lee, from the Campaign for State Education, said: “There are huge dangers attached to putting public money in the hands of private-education providers. That money all of a sudden becomes completely defunct of democratic accountability.”