They have urged the Government not to worsen the schools' cash crisis by awarding inflation-busting pay increases to teachers.
Saxon Spence, education chair of the Association of County Councils, said: "If the Government gives teachers a rise above inflation without helping to cover the costs, then cuts are definitely on the way.
"Every extra pound the Government gives to teachers is another pound taken out of schools."
A nationwide survey of authorities by the Standing Conference of Chief Education Officers has revealed that councils are budgeting for a teachers' pay rise of between 2 and 3.5 per cent. Some however had put in nothing.
Roy Pryke, chairman of the SCCEO, said a pay award of more than 3 per cent would present authorities with real difficulties.
Just 14 out of 59 authorities said they could pass on the extra money ministers say is available for education to the service. Most were talking of cuts to school budgets of between 2 and 4 per cent. Four local authorities predicted they would be higher than 4 per cent.
Special educational needs, home to school transport and pensions were the key areas under pressure with local authority management and administration facing yet another big round of cuts.
Kent, where Mr Pryke is director of education, is looking to cut its administration by 10 per cent. Schools will almost certainly be concerned how this will affect the advice they are given before and after inspections by the Office for Standards in Education.
The cut, if implemented, would also mean that at a time when schools' own budgets are under pressure there would be fewer personnel and finance staff available in County Hall to help them.
Mr Pryke said local authorities were finding it hard to pass on to the service the extra money ministers said was available for education.
"It is clear that because of the pressure on the rest of the local authority budget - social services in particular - that councils are finding it difficult to pass the money on to schools."
Education authorities are already spending hundreds of millions of pounds more on schools this year than the Government has allocated for 199798 and will to have to find more than Pounds 600 million for inescapable costs next year.
The phased increase in pay that ministers gave teachers this year will continue to hit budgets next year with LEAs having to find Pounds 75m to cover the second part of the 3.5 per cent pay deal.
Thousands more children will start school this year at a cost of Pounds 136 million. Other unavoidable new costs such as special educational needs, safer school buses, security and administering nursery vouchers will add Pounds 399m.
"Unfortunately the situation is looking bleak," said Mrs Spence. "Once again the Government's under-funding is going to mean fewer books and supplies for children, cold draughty classrooms and bigger class sizes."
The ACC is to make representations to Gillian Shephard, the Education and Employment Secretary, about its concerns over pay. Mrs Spence said: "The Government has to realise that it cannot go bleating on about constraining public sector pay and then imposing excessive pay increases. If it does not include additional money in the education settlement for pay, it should not impose an unaffordable pay rise."