Bam Construct is understood to have made the shortlist for every region it went for, so far losing out in the first two contests for the Midlands and north east.
Kier is also hoping for its first success after making the final two in three regional bid races worth together around £100m.
Wates wants to build on its early success in the first Midlands batch of schools after taking a place in the last two in the east of England region.
Contractor Balfour Beatty is conspicuously absent from the shortlisting, given it previous track record in the education sector.
School bidders table
South: Bidders Kier and Interserve
London: Bidders Bam Construct and Kier
East: Bidders Kier and Wates
Midlands 1: Winner Wates, £36m
Midlands 2 (Coventry): Bam Construct and Kier
North East: Winner Sir Robert McAlpine, £64m
North West 1: Bidders Bam Construct and Carillion
North West 2: Bidders Bam Construct and Willmott Dixon