Education secretary Alan Johnson, who is favourite to succeed John Prescott, said at the weekend that private schools would have to do more than "lend out their playing fields" to justify the estimated £100m a year in tax breaks they get as a result of their charitable status.
"It's about opening up their science labs, lending their teachers to the state sector, sponsoring academies and forming trusts," he said.
But Dagenham MP Jon Cruddas, an outsider for the deputy leadership, said during that he could see no reason why private schools should be charities at all.
He said: "I don't want to engage in a piece of class warfare, but I have never understood the idea of charitable status for these schools. I would get rid of it and invest the money in the public sector."
Pat Langham, president of the Girls' Schools Association, said Cruddas was engaging in the "politics of envy" and pointed out that the independent sector was held in high esteem. "To seek to destroy it has no real logic other than the denial of something very valuable," she said. "The sector gives back three times what it receives in tax breaks."
Shadow education secretary David Willetts also criticised Johnson and Cruddas.
"Because they are running for the deputy leadership they are making clumsy threats about independent schools' charity status," he said. "Instead we would be far more positive, encouraging independent schools and maintained schools to come together through creating new academies."
Earlier this year draft guidance issued by the Charity Commission on what private schools would have to do to meet the public benefit test brought by the new Charities Act said that the benefit private schools provide must be educational, but that it could include lending their playing fields to local state school sports teams.
The draft guidance also said that one principle of public benefit was that "people on low incomes must be able to benefit."