The move is a blow to the two consortia left in the running: Shape and Bouygues Partnership.
The winning builder was due to be selected in July, but the council said it is now aiming for a date in October, after the Tories have announced the spending review.
Building work on the first two schools is still timetabled to start next year. But there is greater uncertainty about the future of 10 other schools building projects due to start in 2012.
Councillor John Ireland, who is in charge of education in the city, said: “We’ll continue to plan for building work at King Richard and Priory.”
“As for the later phases going ahead, we hope for the best but plan for the worst.”
The project ran into trouble last year after the council reviewed its procurement strategy due to a lack of interest from the computing specialists for the ICT part of the programme.
As a result builders and ICT companies were asked to bid separately to raise competition.