Sir Robert McAlpine, Balfour Beatty, Carillion, Kier, Bam Construction and Wates are fighting it out for the next wave of academy projects, worth more than £70 million, Construction News understands.
Another four of the academies are now down to shortlists of two, with major contractors teaming up with architects for each bid.
Balfour Beatty is part of a joint bid with architects Aedas, while Wates has teamed up with Bond Brian and ACP.
Carillion is bidding with Jestico + Whiles and Bam is with Nicholas Hare.
Kier is also bidding with Nicholas Hare while Sir Robert McAlpine has paired up with Cube architecture.
And Bam Construct and Kier are set to fight it out for the £9m Bradford Science Academy.
Kier is partnered with Haliday Clark and Bam with Taylor Young.
Bam Construct last week became the first contractor to win one of the 28 academy schemes worth a total of £250m.
It secured the £11.7m Kersley Academy in Bolton, pipping Balfour Beatty to the job.