They spoke out as councils began setting their budgets and the threat of school budget cuts emerged.
Mr Blunkett has already warned Conservative-controlled Wiltshire that he does not expect its extra pound;10 million to be spent on other services. He said: "I expect every local education authority to ensure all the cash increase is spent on education."
Policy-makers in Wiltshire have voted to put just pound;7.2m of the extra money into the schools budget. They plan to spend pound;1.3m on social services and highways, and the rest on school buses and central education services.
There were protests this week in the county, and David Ball, head of Rowde primary in Devizes, said: "We seem to work in an LEA which makes a virtue out of pinching pennies."
He estimates that in April his 152-pupil school will have just enough money to pay its six staff. The school has only pound;6,000 in reserves.
As a grant-maintained school, Rowde's budget last year was pound;233,000. Mr Ball says that, with cash taken out to cover central services, his new budget will be pound;205,000. Rowde primary will be returning to voluntary-aided status once the opted-out sector is abolished in September.
"It is ironic that we left the LEA because of under-funding and return three years later to find the same problem confronting us,"said Mr Ball.
Peter Chalke, the council leader, said that Wiltshire was actually giving schools a 6 per cent increase and that Mr Blunkett's letter to the authority was politically motivated.
In the London borough of Haringey, the National Union of Teachers claimed that millions of pounds were going missing.
It said that the authority planned to spend 10 per cent below what the Government had said it should on primaries, and 3.6 per cent less on secondary schools. The authority said that the only cuts yet agreed totalled pound;286,000. The budget would be set next week.
In Redbridge, headteachers claimed the council was cutting pound;5m from the education budget.
In a joint statement, the local branches of the National Association of Head Teachers and the Secondary Heads Association, said: "We can only hope that common sense will prevail."
The council replied that budget proposals put forward by the three political parties detailed cuts ranging from pound;400,000 (Conservatives) to pound;1.6m (Liberal Democrats). The pound;5m had been part of a "paper exercise".
Nationally, councils estimate that they need a further pound;70m to fund the 3.6 per cent pay rise for teachers, and are warning of serious problems, particularly for small primary schools.
The Local Government Association is seeking an urgent meeting with Estelle Morris, the schools standards minister in which it will ask for the extra money.