The councils claimed they were already spending Pounds 800m above the Government's estimate of the amount needed for education.
And Alan Parker, education secretary of the Association of Metropolitan Authorities, said the extra cash Gillian Shephard had won would not make good the cuts that have already been made or meet the needs of the extra pupils in the school system.
Tony Travers, senior research fellow at the London School of Economics, said if the Pounds 800m was earmarked all for schools, it would represent a 4 per cent rise on the standard spending assessment - the Government estimate of what needs to be spent on a service.
He added: "That seems very generous. However it does suggest that the rest of local government spending will be squeezed hard. If this is the case then it will be difficult to prevent the money being used to fund the fire brigade or other local government costs."
It has been suggested the money should be somehow ring-fenced to make sure LEAs did not divert it elsewhere, but it is unclear how this could be done. One way would be to put some of the Pounds 800m into Grants for Education Support and Training, but that would only make a small difference.
The Government believes the extra money will go towards funding the teachers' pay rise. However, the figure recommended by the School Teachers' Review Body will not be known until its report is published at the end of January. The STRB is still waiting for a letter setting out the level of affordability determined by the Budget.
Robin Squire, education junior minister, said in the House of Commons that although the past year had been tight most councils were able to fund the teachers' pay award.
Don Foster, the Liberal Democrat education spokesman, said Pounds 1.3 billion was needed simply to keep education provision at a standstill, not including inflation and the teachers' pay award. Mr Squire replied there was still scope for councils to make further "efficiency savings".
David Blunkett, Labour education and employment spokesman, told the Association of County Councils at its conference this week: "Last year the Tories cut Pounds 500m from the education budget. Thousands of teachers have been made redundant.
"The Tories are not oblivious to the damage inflicted on schools as a result of last year's financial settlement. Mrs Shephard herself has warned Cabinet colleagues that 'insufficient resources now threaten the provision of education in the state school sector including GM schools'."