They remained defiant as Environment Secretary John Gummer gave them until May 2 to appeal against their budgets being capped.
Devon, Gloucestershire, Shropshire, Somerset, Barnsley, Newcastle and Sheffield want to spend a combined Pounds 26.9 million above Government limits and have predicted widespread cuts and job losses - particularly in education - if not allowed to do so.
New research for The TES by Professor Alan Smithers at Manchester University's Centre for Education and Employment (TES, April 7) suggests more than 14, 000 teaching jobs may be lost by August 1996 because of Government curbs on school spending.
This figure is far higher than the 10,000 suggested by Education Secretary Gillian Shephard to Cabinet colleagues last autumn.
Sue Davis, council leader in Shropshire, said: "We cannot go on just giving in to the Government every year. For the sake of all those people across the county who have given us such tremendous support we are determined to pursue our strong case for increased funding.
Shropshire, which spends Pounds 3,000 per secondary pupil - the highest in the country - wants to exceed its cap by Pounds 6 million.
Sue Davis added: "The consequences have already been spelled out clearly enough for school governors and teachers to be unanimously behind the campaign to fight the worst of what they see as Government-imposed cuts."
In Devon, where the authority spends below the national average on each primary and secondary pupil, the authority wants to spend an extra Pounds 4.4 million.
Richard Clark, its chief executive, said: "The extra Pounds 4.4 million above the provisional capping limit would be spent solely in schools to offset an Pounds 8.5 million budget reduction. We hope that Mr Gummer will accept our reasoned, objective arguments and support Devon's education community."
Gloucestershire, whose spending is also below the national average, had agreed a budget Pounds 4 million above the capping limit after the Liberal Democrat and Conservative groups outvoted Labour's plans to spend Pounds 10 million over the limit.
John Braithwaite, deputy chief education officer, said: "We are looking at cuts of about Pounds 3 million in school budgets. This could mean as many as 100 teachers' jobs to go."
Somerset, again with below-average spending, has asked to spend an additional Pounds 2.6 million - an increase of 1 per cent - in an attempt to help schools.
Chris Clarke, leader of the council, said: "This is a very reasonable request. It is one we expect to win on appeal because we have an exceptionally strong case and we have the support of the people of Somerset."
Sheffield, Barnsley and Newcastle were also preparing to appeal.
In Newcastle, where the authority spends above the national average on its pupils, councillors set a budget of Pounds 2.32 million higher than the Government limit.