The summer school, run by Warwick University as part of its national academy for gifted and talented youth, offers 900 places for courses at five universities.

Last week, only 255 applications had been received. Last year’s pilot was free but this summer’s courses will cost up to pound;1,000.

Geoff Lucas, general secretary of the independent schools’ Headmasters’ and Headmistresses’ Conference, said he was contacted by academy representatives in mid-May, asking for addresses for his member schools and those of the Girls’ Schools’ Association.

“It seems a bit of an afterthought,” he said. “Maybe they left us out because they wanted as broad an intake as possible, and then couldn’t make up numbers.”

The decision was also taken to extend the initial, May 30 application deadline by a month.

Educationists have condemned the academy for trying to bring in pupils from privileged backgrounds.

John Bangs, head of education for the National Union of Teachers, said:

“It’s unbelievable. If you agree with extra support for kids from socially deprived areas, it’s daft to have private schools in there.”

Figures for independent-school applications to the summer academy are not yet available. But heads are sceptical about the scheme’s appeal. Graham Able, head of Dulwich College, said: “We have a lot of intelligent boys.

But they get sufficient intellectual stimulation during term. Visiting museums, galleries and theatres would probably better complement what they do at school.”

Peter Dunn, of the national academy, said: “We’ve been sending out a range of mailings. Obviously we felt that encouraging this particular group would be a good thing to do. It was just part of a campaign to keep people informed.”

The Department for Education and Skills stressed that all pupils would benefit from the summer scheme, regardless of background.

A spokesperson said: “The academy targets all potential audiences, including independent schools. But it wants to ensure no one is prevented from joining because of financial reasons.”