Costain said it contacted Mouchel's board on 2 December, and the approach was turned down on 6 December.
The offer comprised 0.5135 new Costain shares for each Mouchel share, valuing Mouchel shares at 105.8p.
Mouchel had already said it had received a preliminary takeover offer, but did not name the interested party.
Costain said it was now uncertain whether it would make a formal bid.
Mouchel's operations include half-ownership of Enterprise Mouchel, which has a contract to grit roads in south London.