The firm is understood to be favourite to take the job ahead of rival bidders Balfour Beatty Rail and Costain.
Sources close to the project said a contract is set to be signed in the next few weeks as Vinci enters into final negotiations with Network Rail, which is responsible for delivering Crossrail projects on the existing rail network.
New station buildings are required at Acton Main Line, Southall, West Ealing and Hayes & Harlington. The station at Ealing Broadway will also undergo extensive renovation including an enlarged ticket hall, new footbridge, new lifts and pedestrianised forecourt.
New lifts and platform extensions will be installed at West Drayton station, while Maidenhead station will benefit from new lifts, a new platform and platform extensions.
A site compound will also be built behind Maidenhead station which will be used for six stabling sidings and staff accommodation for drivers and cleaners.
Platform extensions will be needed at Langley, Burnham, Slough and Iver. Minor architectural work including new signage will be undertaken at Taplow and Hanwell.
In total, Network Rail will manage the upgrade of 43 miles of track and 27 surface stations from Maidenhead in the west to Abbey Wood and Shenfield in the east for the Crossrail project.
Balfour Beatty Rail has already picked up the £130m shorter 2-mile long South East spur which includes track upgrades and a new Abbey Wood terminus station (below).
Network Rail is also inviting tenders for the north east surface section between Stratford in east London and Shenfield in Essex.
Four firms Balfour Beatty, Costain, Hochtief and VolkerFitzpatrick have been asked to submit bids for a £60m package of station improvements and extensions.