John Major privately wants the LSE to take over the site, but has told colleagues that he could not support it if it cost more money.
Michael Portillo, the Chief Secretary to the Treasury, is expected to veto the scheme.
'It's a good idea, but there is no knowing how much it will cost in the long run. If the Treasury is serious about cutting expenditure, it can't possibly allow it to go through,' one minister said.
The LSE has been campaigning vigorously at the Commons, arguing that the sale of its own central London site would make the cost of the move more modest.
Close friends of Mr Major have disclosed that, for months, the Prime Minister has been keeping a close watch on the LSE's bid. A contract for a hotel was signed with a Japanese firm by the London Residuary Body, which took over the GLC's assets.
Mr Major remained in favour of the LSE taking over the building, which could revitalise the South Bank area, encompassing concert halls, galleries and the National Theatre. But he has thrown his weight behind Mr Portillo's demands for cuts in ministerial spending bids.
Mr Portillo will be given the support of his colleagues at a meeting of the Cabinet to review the global spending figures at the end of the month, after the recess of Parliament.
That will act as the go-ahead for a tough round of bilateral negotiations through summer. Mr Portillo's main targets are the road programme being defended by John MacGregor, the Secretary of State for Transport; the housing programme under Michael Howard, the Secretary of State for the Environment; and the defence budget under Malcolm Rifkind, the Secretary of State for Defence.
He is also seeking savings in the social security budget from Peter Lilley, the Secretary of State for Social Security, with employment benefit under scrutiny.
But ministers are privately expressing surprise that in spite of warnings about deep cuts in their bids, there has been little 'blood on the carpet' at the Treasury. Mr Portillo was fought off by the environment department when he tried to cut by half the 20 cities that will secure extra funds for urban regeneration under the City Challenge scheme.
He is insisting that departments stick to the base line for spending agreed in previous years. But spending by departments such as social security is demand-led by unemployment. That is why some leading Tories say that social security benefits must be reined back when unemployment starts to fall.Reuse content