The Enquirer understands that McLaren was given the good news this week.
The club is hoping to start work shortly but is facing a legal challenge from local residents over the plans.
The Elite Training Facility and Community Sports Complex will be built on the Warren Farm site in Southall and is due to be ready ahead of the 2015/16 season.
The training centre will provide: grass pitches for the first team and academy, an indoor 3G pitch, an outdoor 3G artificial pitch, extensive gym, sports science and rehabilitation areas, Academy classrooms, a large number of changing rooms, media, catering and grounds maintenance facilities.
Populous is architect for the scheme.
McLaren’s win comes after its successful delivery of Tottenham Hotspur’s new £45m training ground in Enfield.
The contractor is also working on the first phase of Spurs’ stadium redevelopment plans.
McLaren declined to comment.