It is thought that the Stock Exchange asked Costain to clarify the situation after a rise in the company's share price this week.
Analysts said BICC's civil engineering arm, Balfour Beatty and Amec, were also likely takeover candidates.
A Costain statement said: "The board confirms that it is holding discussions with a number of parties both in relation to the possible sale of its remaining US coal mining assets and in relation to the Costain group as a whole."
Last month Costain sold a substantial part of its US interests in Louisiana to a subsidiary of Philipp Holzmann for £63.5m. The rest of its coal operations, in Kentucky, is for sale at what Costain said would be "a substantial discount" to book value.
The company, which has made pre-tax losses since 1991 of £219m, has already already been forced to sell its British residential and commercial property interests to reduce large debts. The Australian coal division has also gone.
P&O, which owns Bovis construction group, has been linked as a potential bidder for some time. One analyst said yesterday that P&O had confirmed it was "potentially" interested.
However, Nick Wilson, of Nomura, was more cautious about the suitability of a takeover, emphasising the differences in the two companies' construction businesses. P&O's construction activities are more internationally-orientated while Costain is heavily British-based. Furthermore, Bovis prefers to tender for contracts on a fixed-fee basis while Costain typically tenders for an entire construction contract, including materials. P&O, and Hanson, declined to comment.
Michael Foster, of Greig Middleton, thought a takeover unlikely in the short term. Although a bidder would get a discount for the company's equity, it may face increased payments on delayed contracts.
Costain faces yet-to-be-realised tax losses on the disposal of its US operations. But it faces an easier tax burden in Britain, having benefited from the disposal of Costain Homes and its associated losses.
Costain's market value is put at about £100m. The shares, which began the week at 18.75p, closed up 1p yesterday at 23.5p.