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It is understood that Morgan Sindall and Balfour Beatty, mentioned by analysts as rival bidders for the contractor, have ruled themselves out.

Galliford Try, another potential bidder, is also not thought currently to be looking at the firm, which would leave the way clear for Carillion or a private equity bidder.

McAlpine this week rejected a 拢584m approach from Carillion insisting it 鈥渕aterially undervalued鈥� the company.

McAlpine also revealed it had rejected a 拢574m bid from Carillion in August. McAlpine鈥檚 market capitalisation is about 拢565尘.

Carillion鈥檚 shares fell 7% after the announcement, an indication that the City believes it will continue to pursue McAlpine. The takeover talks began after Alfred McAlpine disclosed a plan to split the business into two: business and infrastructure services, and project services.

One City source said: 鈥淎n offer of about 600p is likely to seal a deal. That price gives a certainty over the potential of the demerger plan.鈥�

A Carillion spokesperson said: 鈥淎lfred McAlpine would represent an attractive acquisition and a combination of the two companies would present an excellent strategic fit.鈥�

However, City sources have raised questions over Carillion鈥檚 ability to integrate the firm so soon after its acquisition of Mowlem, which involved a 拢135m hit on problem contracts.

McAlpine insisted its 鈥渟trong recent results鈥� vindicated the decision to reject the offer from Carillion.