The Heseltine scheme would compel education to compete against housing, roads and social services for funds for capital projects. Those that had attracted "matching" private finance will get priority.
Civil servants in regional government offices will choose, for example, between repairing a leaking school roof, introducing a traffic-calming scheme and building an old people's home.
The idea is being fiercely opposed by Gillian Shephard, Secretary of State for Education, and by Sir George Young, Secretary of State for Transport. They would lose control over some capital projects in their areas.
Mrs Shephard fears the Heseltine programme - known as "Capital Challenge" - will divert money into glamorous building projects, which are more likely to excite the private sector, while school roofs leak and classrooms crumble.
She has already announced a rival scheme, worth pounds 40m, and has sent letters to schools inviting them to bid for funds to improve school buildings.
It is the second time in a week that Mr Heseltine has been revealed in a dispute with fellow ministers. Last week a leaked letter revealed an angry dispute between him and Ian Lang, President of the Board of Trade, over plans to strip up to nine million workers in small firms of their rights to contest unfair dismissals.
Mr Lang described the proposals as "immensely controversial".
The deputy Prime Minister's critics see the latest row over the "Capital Challenge" programme, which is also backed by John Gummer, Secretary of State for the Environment, as a further indication that he has lost his political touch. The scheme, they say, is a potential vote-loser.
At present, capital grants are allocated to local councils by four separate government departments - education, transport, housing and social services. Under the Heseltine scheme, local authorities would send all their bids to one of the 12 regional offices, and the civil servants would choose.
For the first year, only pounds 300m out of the annual pounds 3.5bn local council spending on capital projects would be allocated in this way. But the sum would rise rapidly and the long-term aim is to bring all capital funding into the scheme. It would be an enormous shift of power away from Whitehall spending ministries to regional offices.
Of the spending ministers, only Stephen Dorrell, Secretary of State for Health, is giving enthusiastic backing. As a Treasury minister he was once responsible for the private finance initiative.Local councils, which will lose the money going to Mr Heseltine's scheme, fear that it may leave them without enough to provide classrooms for all children.
David Whitbread, education under-secretary at the Association of County Councils, said: "The projects which attract private money in a challenge scheme might not be the most essential. You might get a community centre on a secondary school site, which would generate some income and involve partnership with voluntary bodies. The school would get new facilities, but there might be another school down the road where the roof was about to blow off or one where there were 40 children in a class and no money for an extra classroom. It's a distortion of priorities."
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