Amec had been in pole position for the project after being recommended by the council's in-house and professional team (30 April), but a reassessment is now taking place.
A source within the team said the decision to appoint unnamed external advisers had been made by council chief executive Howard Bernstein.
Bernstein was expected to appoint the contractor recommended by the in-house team and announce a decision by 6 May. This date has now been put back to the end of May.
The council source said: "Howard is doing everything he can to ensure we make the right decision.
"He is scrutinising it [the recommendation] thoroughly and he has brought in external advisers to take a second look.
The council needs to be careful of allegations of favouritism in everything it does." The three front-runners for the scheme were Amec, Bovis and Laing; Mowlem and Taylor Woodrow, failed to make the shortlist.