The company was last week named as construction manager for the job, which is due to be finished by May.
The firm was one of four contractors, which included Turner Construction, Amec and Tully, chosen to work on the site in the wake of the terrorist attacks on 11 September.
The move follows speculation at the end of last year that rival Bechtel would take over.
In a statement, Bovis Lend Lease said New York City’s Department of Design and Construction, the client for the project, wanted a more traditional project management structure, rather than sharing the work between four firms. Amec and Tully will remain as subcontractors. Turner has finished its work.
Paul Ashlin, the Bovis senior vice-president who is overseeing the operation, said the work was running ahead of schedule.
He said: “Initially we thought the operation would take a year – we now think we can finish it by the end of May.
“It’s gone faster than anybody anticipated. We believe we have organised the thing well. We have received tremendous co-operation from city officials and site owners, the Port Authority.”
Ashlin said the firm’s priority was to stabilise the 16-acre site below the ground, known as “the bath tub”. This will be done by inserting steel strands through three of the four walls of the bath tub into nearby rock.
The firm is planning to build a prefabricated bridge across the site, to allow articulated lorries to pick up debris.
The four contractors have removed 1 billion tonnes of debris since September.
Ashlin said the team was also excavating the area between the bath tub and a nearby highway.
He added that the next big job, ahead of the redevelopment of the site, was the rebuilding of the underground transport interchange that links Manhattan with New Jersey.