Kier and Bam Construct are on the hunt for two of the jobs, which have come from the pipeline of academy schemes.
Eventually 28 schemes worth £250m will be released in the North, and a further 43 schemes worth £550m will be allocated for the south.
In Wiltshire, Kier and Sir Robert McAlpine are reported to have made the final shortlist for the £14m Sarum Academy.
While Balfour Beatty and Wates are understood to be battling it out for the Thetford Academy in Norfolk, which is the biggest of the next wave of jobs at an estimated build cost of £26m.
Bam Construct is understood to be up against Carillion for the £22m Aylesbury Vale Academy in Buckinghamshire.
While in the northern region Bam Construct is also hoping to score another success in a race against Kier for the Bradford Science Academy.
Bam recently won the first of the new academy projects to come out in Bolton, when it secured a £12m contract for Kersley Academy.