The Enquirer understands Carillion has beaten off competition from Morgan Sindall and Skanska for the Stockley Viaduct in Hillingdon near the existing flyover.
Jacobs Engineering has designed the flyover to provide a separate route for trains crossing the main line and to enable Crossrail services to operate to Heathrow.
One bidder said: “Nothing has been made official yet but the word doing the rounds is that Carillion has got this and will start on site shortly.”