The firm is close to completing a £19m science block at the private school beside the bank of the river Thames.
It is now set to take the bigger contract for the main general teaching building, the Enquirer understands.
Mace is finalising design work following detailed planning permission in the Summer and is expected to start the project early next year.
The new building will contain most of the teaching and learning, and some of the administrative functions of the senior school.
Its demolition and replacement will be the most complex element of the overall school rebuilding programme.
