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This season's European Champions Cup and Challenge Cup quarter-finals will not be played before September after the French government banned all sport for the next four months.
The last eight of the competitions, involving six French teams, were due to take place from 3-5 April.
But they had already been postponed because of the coronavirus pandemic.
Organisers European Professional Club Rugby still want to complete this season's competitions.
Irish sides Leinster and Ulster as well as English teams Saracens, Northampton and Exeter are all due to play in the Champions Cup last eight.
In the Challenge Cup Welsh clubs Scarlets and Dragons, Scottish side Edinburgh and Premiership teams Leicester and Bristol have all qualified for the knockout stages.
The finals of the two competitions had originally been scheduled to take place in Marseille next month.
If the competitions are now completed in September or October, it would have a knock-on effect on the 2020-21 tournaments, which are due to begin on 16 October.
French Prime Minister Edouard Philippe said on Tuesday that the 2019-20 sporting season was over as he announced plans to ease France's coronavirus lockdown on 11 May.
Earlier on Thursday it was announced that French rugby union's top two divisions - the Top 14 and Pro D2 - would be abandoned for the season.
The decision could set a precedent for the English Premiership, which is working towards restarting in July, and the Pro14.
Bordeaux Begles are currently leading the Top 14, eight points clear of Lyon.
The league winners are usually decided via semi-finals and finals, but the league's organisers are yet to announce whether any club will be named as champions.
Chris Jones, BBC rugby union correspondent
Unlike their English counterparts, the French clubs do not want to resort to playing without crowds, and with mass gatherings banned until September, the focus will now be on trying to start the 2020-21 season on time.
The Premiership and Pro14 are in a difficult situation, and will be feeling the pressure. While the clubs are in desperate need of the TV money that would come with behind-closed-doors matches, there are numerous hurdles to overcome to make this happen.
Already this week, Glasgow coach Dave Rennie has pointed out the strain it will put on next season's domestic scene if the current campaigns do not get finished until the end of the summer.
