Calverley will now stay until the start of next year, by which time the group's construction operations should be revamped.
A City analyst said: "If Sturgess sorts out the construction side I'd have thought he would have a good chance of succeeding Calverley."
Sturgess, who was UK building managing director at Skanska for five years until his resignation last year, refused to comment on the speculation. He said: "My job is very straightforward. I've got a full-time job."
Sturgess said the company would concentrate on long-term commercial work and jobs in the regulated market.
Galliford Try chairman Tony Palmer said of the appointment: "Andy brings to us the experience of managing one of the UK's leading building contractors, and in particular has extensive expertise in working for the public and regulated sectors as well as for commercial clients, much of it through partnerships and framework agreements."
In a further move, Chris King, managing director of Galliford UK, is to become business development director of the group's construction division.