The election will now be held again. A union source said Davies had decided to run against Coulter, 42, the most senior female union official in Britain. The source said voting was due to begin this week and would last for a week.
Two other candidates are standing in addition to Coulter and Davies: Bob Stokes, of Fareham in Hampshire, and another female official called Giovanni Holt.
Coulter's election was disallowed by the certification officer, the GMB's ombudsman, after Stokes complained that he had been unfairly excluded from the election. Bob Stokes said he had been told he could not stand because he was too old.
The union lost an appeal against the certification officer's ruling and has now been forced to hold a new poll of its 70,000 members at a cost of about £250,000.
The union source said Davies, who has a reputation as a conciliator, was likely to remain in control of construction if he won the vote.
He said: "Davies is likely not to act like a traditional deputy general secretary who is in bed with the Labour Party. It is thought he will be more hands-on and in close contact with the big industries such as construction."
The source added that it would be a relief for the union when the result of the election came in. He said: "The union has been rocked by the controversy – we need a period of stability now."