The Enquirer understands that the Laing O’Rourke / Bachy Soletanche / Costain joint venture is in pole position for the job.
The contract is part of a £300m revamp of the underground which will see it link up with a new Bond Street Crossrail station.
A formal announcement on a winner is expected by the end of the summer.
The other groups in the race are Balfour Beatty and a Taylor Woodrow / BAM Nuttall joint venture.
A source close to the talks told the Enquirer: “This has been a really tough fight but it looks like the O’Rourke team has edged ahead.
“Nothing’s official yet but the feeling is the job is going their way.”
Main construction is due to begin later this year with the redevelopment of the station completed in early 2017.
The winning contractor will deliver an expansion of Bond Street Tube station as well as a step-free link between the tube station and Crossrail Bond Street station.
A separate contract will be let by Crossrail in due course for the construction of the new Crossrail station at Bond Street which will connect to the Tube station to form one overall station.
A spokesperson for Transport for London said: “By 2017, Bond Street station will be served by both London Underground and Crossrail and will be the gateway to the West End.
“TfL plans to award the contract for the major redevelopment of Bond Street Tube station this summer as previously announced.”
Laing O’Rourke declined to comment.