The move comes after the French sports ministry recommended postponing January’s home and away fixtures.
The European Professional Club Rugby (EPCR), which runs the tournaments announced the halt after talks with the Ligue Nationale de Rugby, which organises the Top 14.
"Following a directive from the French authorities that the participation of Top 14 clubs in the Champions Cup and Challenge Cup at the current time constitutes too great a public health risk, the EPCR has no choice but to announce that the 2020/21 tournaments are temporarily suspended," it said in a statement.
The Champions Cup was set to resume on Friday with Northampton Saints taking on Leinster while Toulon were scheduled to host Scarlets.
Britain has recorded more than 3 million confirmed Covid-19 cases and more than 81,000 deaths. France has registered nearly 70,000 deaths since January 2020.
Break
Following the announcement, Premiership Rugby, which oversees English top flight rugby union in England, said it would take a two-week break.
The campaign will resume on 29 January when Bristol Bears host Bath.
"The welfare of everyone involved in Premiership Rugby is crucial to us and this break in the season gives us the chance to hand some much needed rest to our hard-working players, management, staff and match officials," Premiership Rugby chief executive Darren Childs said.
The Six Nations tournament, featuring France, England, Scotland, Wales, Ireland and Italy, is due to begin on 6 February.
Bernard Laporte, France’s top rugby unon administrator, said he remained confident the 2021 competition would proceed despite the coronavirus pandemic.
The 2020 Six Nations tournament was halted between March and October after national lockdowns were imposed to stem the spread of the disease.
