The track between Rayners Lane and Uxbridge has been shut for several weeks after worn-down wheels and motors led to a shortage of trains.
Leaf fall caused by recent weather conditions has been blamed for damaging the fleet, with some trains needing to be taken out of service and repaired.
This has also resulted in delays on other parts of the Piccadilly line over the past two months.
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Transport bosses have now said that the closed section of the line is on track to be reopened later this month, although an exact date is yet to be confirmed.
Full service on the rest of the line will also be restored once this happens.
Richard Jones, director of asset performance delivery for TfL, has apologised to Piccadilly line customers for the disruption.
He said: “The trains, which are among the oldest on the network, have sustained damage to the wheels and motors due to recent weather conditions.
“We have not operated Piccadilly line services between Rayners Lane and Uxbridge for some time due to the rail conditions in the area and the number of trains we have available for service.
“Essential repair work has had to be completed to ensure that the trains continue to operate safely and reliably ahead of the introduction of new Piccadilly line trains.
“The repair work is progressing well and we expect to be able to restore a full Piccadilly line service in late January.”
Piccadilly line passengers can expect more disruption in the coming months while TfL prepares for the introduction of new walk-through, air-conditioned trains.
There will be 13 weekends of partial closures along the line until the end of June this year.
