David Blunkett, the Education and Employment Secretary, warned council leaders this week that the Government would seriously consider bypassing them altogether if in spending the cash they did not keep to the spirit of what he described as a "good faith gesture" by the Government.
Ministers are under pressure to introduce a national funding formula to put money directly into school budgets, an idea considered by the Conservatives.
Officials at both the Department for Education and Employment and the Funding Agency for Schools have been working on such a formula, but the idea has so far been resisted by Labour.
Its introduction was threatened by Mr Blunkett, who told the Council of Local Education Authorities' annual conference in Bristol: "We need to remind leaders of councils and directors of finance that we are under pressure for a national funding formula."
In a later briefing to journalists he said: "We have resisted (the NFF) but if education authorities are not able to translate the resources provided into schools we would have to re-examine our options."
He told local authority leaders: "There are many people arguing for a national funding formula for schools. I still believe that there needs to be a role for LEAs in this area, but with such a role comes a clear responsibility to deliver. There is no better opportunity for councils to persuade those who favour a national funding formula that they are able to deliver on education. "
He made clear that he expected the School Teachers' Review Body to limit salary awards for the profession to inflation, saying the extra money was for raising standards "not to mop up pay increases".
Local government is also pressing for an inflation-linked pay rise, which it believes would cost an extra Pounds 400 million. This would amount to almost half of the additional money the Chancellor gave to schools in England in his Budget (Pounds 835m).
Mr Blunkett said local authorities would need to balance carefully what needed to be spent to retain teachers and what should actually go into schools.
He has written to every education authority saying he expected the extra money to be used for the benefit of schools. "The extra money was an indication of good faith in terms of us recognising we were facing a calamity - the prospect of mass redundancy and reduction in investment in reducing class sizes. "