David Cameron, the new Conservative leader, has promised Tony Blair that the Tories will help him to push through controversial school reforms.

“With our support you know there is no danger of losing these education reforms in a parliamentary vote, so you can afford to be as bold as you want to be,” he told the Prime Minister.

Mr Cameron used his first appearance at Prime Minister’s Question Time to pledge support for the controversial education Bill, which has prompted threats of a Labour backbench rebellion and talk of defeat.

Up until Tuesday, when the 39-year-old won the leadership contest, Mr Cameron had been the Tories’ education spokesman.

He has promised to bring an end to “Punch and Judy politics”. But he said the Conservatives would continue to push for all schools to have control over their admissions, a proposal which has been flatly rejected by Mr Blair.

Backbench Labour MPs are threatening to revolt over the education Bill, expected in the new year, particularly over plans for new independent “trust schools”.

Last week Martin Salter, Labour MP for Reading West, resigned as the parliamentary private secretary to Jacqui Smith, the schools minister, because of “serious reservations” about the recent education white paper that forms the basis of the Bill.

MPs this week heard criticism of the white paper from all six of the main teachers’ unions and England’s children’s commissioner.

The unions unanimously called for the Government to drop its plans to let schools become governed by independent trusts, which could be set up by faith groups, community organisations and businesses.

Steve Sinnott, general secretary of the National Union of Teachers, said international evidence indicated the plans would be detrimental to community cohesion and pupils from disadvantaged backgrounds.

Headteachers’ associations said their members had shown little interest in becoming trust schools as there was no real incentive.

Earlier Professor Al Aynsley Green, the children’s commissioner for England, told MPs that he was concerned the paper’s proposals would not sufficiently help disabled or bullied pupils and might create “very isolationalist schools”.

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