The latest delay would put the scheme nearly three years behind the original planned opening of December 2018.
Crossrail bosses are holding a board meeting on Thursday when more details about opening dates will be confirmed.
But TfL Commissioner Mike Brown confirmed on Monday that the transport body was working towards the “pessimistic but pragmatic” date of between September and December 2021.
Brown was speaking to the London Assembly’s budget and performance committee.
Previous indications from Crossrail were that the line would “open as soon as practically possible in 2021.”
That latest delayed date came last November when the project also announced it needed another £400m – £650m in additional funding.
Brown confirmed an earlier Enquirer story that M&E systems across a lot of the job were in disarray.
He said: “If you wandered around the stations, as I did at this time last year, these stations looked as though they were complete.
“I went back a few months later and tiles had to be taken away, platforms had to be re-opened up because of some of the wiring of the system, some of the processes and equipment which were supposed to have been installed simply weren’t there.”
A TfL spokesperson said: “The Crossrail team continues to make progress completing the railway and is moving forward with the complex testing of the signalling and train systems so that the railway can be handed over safely and reliably for passenger service.
“As Crossrail Ltd previously announced, the Elizabeth line will open as soon as practically possible in 2021.
“As part of our annual business planning process, we have made some prudent assumptions including that the central section of the railway could open in autumn 2021, but continue to support Crossrail Ltd in delivering the railway as soon as possible.
“Crossrail Ltd continue to refine their delivery schedule and will provide an update in the coming weeks.”