The firm is up against Balfour Beatty, Hochtief and VolkerFitzpatrick for the major station improvements and extensions package, which are being managed by Network Rail.
Sources close to Network Rail said a decision is due to be announced shortly.
The winner of the bid race will build new stations at Romford and Ilford, as well as improvements at stations including Forest Gate, Goodmayes, Harold Wood, Gidea Park, Chadwell Heath and Brentwood.
Platform extensions will allow longer, higher capacity trains to run, and new lifts to enable step-free access will be installed at many stations along the route.
The detailed design phase will start shortly with main works starting between late 2014 and 2017.
Other works will include various infrastructure improvements, such as extra train stabling capacity and turnback facilities that will improve the reliability of passenger services.
Under the Crossrail programme, Network Rail is responsible for the upgrade of 43 miles of track in total, as well as improving 27 stations from Maidenhead in the west to Abbey Wood and Shenfield in the east.
Vinci Construction UK won the £100m job to design and upgrade 13 stations in west London and Berkshire, covering significant improvements to stations from Acton Main Line to Maidenhead.
Balfour Beatty took the £100m bundled job to upgrade lines and build Abeey Wood station on the shorter south east Spur of Crossrail