The councils - who call themselves the E7 group in an ironic reference to the G7 group of the world's richest nations - have told Stephen Byers, the schools standards minister, they must have more money.
They say there are increasing numbers of classes with 30 or more pupils - at a time when the Government has committed itself to cut class sizes for every five, six and seven-year-old.
In Derbyshire, almost 65 per cent of primary pupils are taught in classes of 30 or more - an increase of more than 10 per cent in the last two years. Nearly are quarter in classes of 35-plus.
Dave Wilcox, Derbyshire education chair and chair of the E7 group, said: "With new literacy targets being set for councils we need the extra cash to tackle class size to help drive up standards in schools."
The E7 authorities reported rising levels of sickness and stress among headteachers and teaching staff, and said that both they and their schools now had little or no cash reserves.
"We're all experiencing the same kinds of problems in seeing class sizes grow, teachers jobs being lost and a reduction in advisory and curriculum support to schools," said Mr Wilcox.
"But the situation for E7 group members is worse than for most councils because we receive much less cash per pupil than in most other areas. The effects of the funding shortfall have been felt not only in our classrooms but in all areas of education spending. Our administrative costs have been cut to the bone . . . leaving us little flexibility for taking on a bigger role in helping improve school performance."
The E7 group, set up in April 1995, was originally made up of Derbyshire, Hereford and Worcester, Northumberland, North Yorkshire, Staffordshire, Suffolk and Warwickshire.
It has now grown to include the counties of Cheshire, Gloucestershire, Northamptonshire, Dorset, Wiltshire and Somerset. They are among the poorest local authorities in England and the amount they receive from the Government for each primary and secondary school pupil is below the county average.
The authorities are concerned now about the effect on other local authority services whose budgets are being raided to provide extra money for education.
"We've all made education spending a priority," said Mr Wilcox. "But it's difficult to maintain this position when the costs to other council services are rising."
Many have now virtually stopped making discretionary awards to post-16 students.