The game was due to take place at the Stade de France on April 4.
France's Ligue de Football Professionnel (LFP), which held talks with both clubs and the French Football Federation (FFF) on Wednesday, has not yet suggested a new date.
The competition is due to be scrapped after this season in order to reduce player workload.
As a result of the final being postponed, the PSG-Metz and Lyon-Nimes league matches will be staged over the weekend of April 4-5, behind closed doors.
The LFP had already confirmed on Tuesday that all Ligue 1 and Ligue 2 matches are to be held without fans present until at least April 15 as part of nationwide measures aimed at curbing the spread of COVID-19.
PSG are due to face Borussia Dortmund in the Champions League last-16 second leg at Parc des Princes on Wednesday, which will also be behind closed doors.
There have been 1,748 confirmed cases of the virus in France, with 33 people killed.