Chief executive Mark Wild told the London Assembly Transport Committee that there are “thousands of hours of construction work still to be done.”
He said construction work on the stations is now scheduled to be completed this summer before full fit-out work and testing of the line gets underway.
The scheme is currently £2.5bn over budget and a new opening “window” will not be outlined until next month.
Wild said: “By the end of July we are looking for major construction work and wiring testing at the stations to be completed.
“We will then hope to have an opening window by the end of quarter one, but I have no idea how wide that window will be.”
The committee heard how the previous Crossrail management team had bungled initial system integration and testing on the route which was started too early alongside live construction work.
Wild added that the previous executive team had “been overwhelmed by the complexity of the task” of building Crossrail.
A number of former executives were paid hundreds of thousands in performance bonuses as they continually insisted the project was “on time and on budget”.
Wild took over as Crossrail chief executive in November when he discovered construction work on none of the stations was near completion and extra funds needed were £2.5bn rather than previous estimates of £300m.