The Crossrail Bill is expected to get Royal Assent on 22 July with the first major job on the deal at Tottenham Court Road due to start next January.
But Mr Harris admitted there was a chance it might get held up before the Parliamentary summer recess, meaning it would not be passed until the autumn. He added: “I’m 75 per cent sure that it will get through in this session - I’m doing everything I can.”
Any delay would put back the start of work at Tottenham Court Road since a series of compulsory purchase orders - including one to tear down the London Astoria theatre - are planned for October in order for work to start.
A Crossrail spokesman said the third reading of the Bill was due to take place this week. The next step will be finalising the funding with the Treasury and Department for Transport, which is due in September.
The first major job on Crossrail will involve revamping Tottenham Court Road Underground station in central London and building a new box station for Crossrail trains.
Taylor Woodrow, Balfour Beatty, Costain, Laing O’Rourke and Morgan Est are all believed to be interested in the job, which is expected to be worth up to £250 million.
The work will be managed by London Underground. A tender notice is expected in the Official Journal later this summer.
The job will include new entrances at Dean Street and Charing Cross Road, expansion of the ticket hall, five new lifts to provide step-free access from the ticket hall to all platforms via new tunnelled passageways, and additional escalators.
Preparatory work to divert utilities including water and gas mains has been going on for more than a year. LU said enabling work for the main construction will begin soon.
There are also plans to create a more pleasant environment overground by building wider pavements, better cycling facilities, bus improvements and a new public piazza.
The whole job is expected to take around six-and-a-half years to complete.