The school's governors have invited Allies and Morrison, David Chipperfield Architects, Fielden Clegg Bradley, Nicholas Hare Architects, Porphyrios Associates, Rafael Moneo, Stanton Williams, Dixon Jones and Wright & Wright to bid for the project.
The winner will draw up extensive proposals for 40,000m 2of the school's 18a riverfront site, immediately to the west of Hammersmith Bridge.
The brief demands that these designs work within a masterplan for the whole site drawn up by Patel Taylor in 2003.
Major new buildings are likely to include an assembly hall and chapel, a science centre, a boathouse/pavilion/visitors' facility, a new music building and residential accommodation.
The majority of the school's current buildings were constructed in 1968 using a popular prefabricated system.