The firm is rebuilding Lees Brook Community School on its existing site in Chaddesden in Derby.
Under the intensive 12-month programme, BAM is constructing a three-floor main building for 1,100 pupils aged 11 to 16 and refurbishing an existing sports hall.
BAM hopes to receive planning permission this year to start on site at Hallmoor School in Birmingham and Fountaindale School in Mansfield.
A planning application for Heathlands Primary School in Birmingham is expected soon.
Construction director David Ellis said: “It is fantastic to be getting to work on one of the first school projects through the Priority Schools Building Programme.
“With our sister company BAM Design providing the architecture, services and structural design, we are able to offer an integrated design-and-build team that gets to work on urgent projects like this.”