The opening of the central section between Paddington and Abbey Wood was due to open next summer.
Crossrail confirmed on Thursday that this is now “not achievable.”
A statement said: “A programme of this scale and complexity was already challenging, the impact of Covid-19 has clearly made the existing pressures more acute.
“Due to a pause of physical activity on sites and significant constraints on ongoing work – time has been lost, only some of which can be recovered.”
Construction work stopped due to Covid-19 on March 24 and restarted on June 15. Train testing work restarted earlier on May 30.
It is the latest delay to a project which was originally scheduled to open in December 2018.
Crossrail chiefs have not issued a revised timetable but said “a more comprehensive update will be issued in due course.”
They added: “Work continues to refine and validate the remaining work schedule and associated costs.”
Mark Wild, Chief Executive, Crossrail Ltd, said: “We have a comprehensive plan to complete the railway but existing schedule pressure along with Covid-19 has impacted the programme and time has been lost.
“Further work is being undertaken to finalise our detailed recovery plan which re-sequences the remaining work.
“We are striving to commence intensive operational testing for the Elizabeth line, known as Trial Running, at the earliest opportunity.
“Our focus right now is on completing the remaining infrastructure works so that we can fully test the railway.
“Crossrail will undertake a period of intensive construction activity starting this August across the tunnels, portals and shafts to help recover lost time and complete the remaining construction works for Trial Running as we progress towards this next important phase.”
Crossrail confirmed last month that it was parting company with main contractor Costain/Skanska joint venture at Bond Street and will oversee the completion of work itself.