The initial longlist was rumoured to have included former Mowlem building division boss Steve Bowcott, former Hanson Building Materials Europe chief executive Simon Vivian and current Peterhouse chief executive Alan Robertson.
The names on the shortlist are not known, but Building understands that Robertson has failed to make the cut.
Bowcott left Mowlem to join Amec in 2002. Industry sources said that he would have been a frontrunner to replace Gains if he had remained with Mowlem.
Steve Bowcott is a Mowlem man through and through. If he were offered the job it would be irresistible
Source close to Mowlem
A source close to Mowlem said: "Steve is a Mowlem man through and through. If he were offered the job it would prove irresistible."
Industry sources speculate that Bowcott left Mowlem because he grew tired of waiting for the top job. One source said: "Bowcott might have pushed too hard to get the job, at a time when Gains was not ready to give it up. If he had stayed he would have inherited the job."
Vivian left his role at Hanson in June last year, after his unit was shut down. At the time it was rumoured that he might have been given the chief executive's job at concrete group RMC.
Mowlem declined to comment but has said in the past that Gains' retirement date is his own business.