The Enquirer understands the contractor has signed a Pre-Construction Services Agreement with the developer.
The Westminster scheme has a total development value of £380m and involves a complete remodelling of the existing 29-storey building.
McLaren’s contract is understood to be worth around £130m.
Original redevelopment plans granted for the building in 2020 included a 15-storey extension with Mace understood to have been favourite for the job.
But changes to the office market prompted a complete rethink and revised refurbishment plans.
Construction is expected to start this summer with an initial commitment to early works only by Landsec as the developer seeks to slow delivery down while the commercial market continues to settle.