In a perfect world there would be "no need for independent schools", the headmaster of Eton said at a conference of leading headteachers last week.
Anthony Little, head of arguably the most famous public school in the world, said that a completely state-run school system was what everyone would "wish for".
But speaking at the 100 Group conference in east London, a gathering of the top 100 state and independent heads, Mr Little admitted that he had little "faith" that the country could do without independent schools.
"In the world we all wish for there would be no need for the independent schools system, it would all be state run," he said.
"But I don't have the faith pace (with all due respect to) any government in power that could create an environment where we could do without the independent school sector."
Mr Little, who came from a working class family but attended Eton thanks to an assisted place, was speaking at the conference themed around social mobility.
When asked whether education supports social mobility, the Eton headmaster replied: "Absolutely not. We have a very long way to go - I think there is a huge disparity in social mobility.
"We have a dialogue that's very aware, but we have to give opportunities across the board - we have yet to live up to what we want to be achieving."
Mr Little called on any future government to focus on areas such as why girls do better academically than boys, and questioned the curbing of competition in the state sector.
He said: "One area we need to look at is why girls are outperforming boys. There is genuine work to be done around how we help boys be themselves.
"I think there is also an issue where in some schools competition seems to be dampened. There is a lot more that we can do, but at the moment I don't think education supports social mobility."
Andrew Grant, chairman of the Headmasters' and Headmistresses' Conference, disagreed.
Mr Grant said: "I would say my perfect world would be one where every school was an independent school, as it is the Government who pose threats to schools. The opposite of independent is dependent, and state schools are dependent on the Government for funding, which pulls many other levers.
"They must toe the Government line, which causes compromises. It's dependency that needs to go."
Mr Little's comments followed earlier claims by another leading independent head Barnaby Lenon, head of Harrow School, who said the Government must avoid "dumbing down" in its attempt to tackle problems around social mobility.
Mr Lenon claimed poorer children were being deceived into taking "worthless qualifications" that fail to provide them with the necessary skills in later life, describing them as being akin to "citizens of Weimar, Germany, or Robert Mugabe's Zimbabwe carrying their certificates around in a wheelbarrow".
He said: "If we want the brightest children from our poorest homes to fulfil their potential we must not deceive them with high grades in soft subjects or allow them to believe that going to any old university to read any subject is going to be the path to prosperity, because it's not."
In good company
The 100 Group is a collection of the 50 leading state school heads and 50 independent school heads selected in discussion between Schools Secretary Ed Balls, headmaster of Brighton College Richard Cairns, and Joan Deslandes, head of Kingsford Community School, where the event was staged.
The heads were chosen on the basis of their standing in their sectors, their commitment to furthering co-operation between the independent and state sectors and their public contributions to the education debate.
The comments of Tony Little came just days after the Government gave tentative backing to former health secretary Alan Milburn's report on social mobility.
Despite claims that the Government had accepted the vast majority of the report's findings, a host of recommendations were shelved, such as parental vouchers for failing schools.
In response, a Department for Children, Schools and Families spokesperson said: "In the response to Alan Milburn's report on access to the professions, the Government set out a clear plan to promote the aspirations of all young people and increase social mobility."