The scheme has been design by Wilkinson Eyre architects and will be built at Sebastian Street near the City University Islington campus.
The single stage design and build tender is being hotly contested with Kier, Wates, and Willmott Dixon among the firms to have chased the project.
But sources have told the Enquirer that Bam is now in prime position to secure the project.
City University plans to retain and refurbish two existing buildings as well as fund a new four-storey brick building and new seven-storey glazed building as a focal point for the the law school complex.
AECOM is acting as M&E consultant, with Sweett on board as project cost consultant .
Work is expected to start on the project in August with completion expected in May 2019.