Bam Construct and Bovis will battle it out for the job, which will first see the new Strood Academy co-educational facility built for use by September 2012.
The Enquirer understands ITT bids have to be submitted by mid-August, with a preferred bidder selected by end of September at the latest.
A further two academies, Bishop of Rochester and Gillingham, will then follow from the single procurement process.
The Medway Academies programme is part of the wider transformation of Medway’s portfolio of schools, which will be created by merging five existing schools into three – each with a capacity of 1,500 students.