Two rounds of pool games in both European club competitions - due to be held on the weekends of January 15-17 and 22-24 - have been postponed as authorities react to the impact of a new strain of Covid-19.
The French government had advised - but not ordered - their teams to avoid cross-border matches on Saturday.
An ECPR statement read: "Following a directive from the French authorities that the participation of TOP 14 clubs in the Heineken Champions Cup and Challenge Cup at the current time constitutes too great a public health risk, EPCR has no choice but to announce that the 2020/21 tournaments are temporarily suspended.
"EPCR and the Ligue Nationale de Rugby took part in a meeting by audio conference last week with representatives of the French ministries of the Interior, Health and Sport, as well as of the President's office.
"Notable updates to EPCR's COVID-19 protocols were presented to the French authorities, including the addition of PCR tests no earlier than three days before matches in the tournaments, conforming to French government guidelines and supplementing the existing, meticulous contact tracing measures put in place with a view to limiting the risk of transmission.
"Against the backdrop of the recent detection of a new strain of coronavirus, the French government has directed that French clubs postpone their participation in EPCR's tournaments for the month of January, both for matches scheduled in France and for those due to be played in the United Kingdom and Ireland.
"On the basis of this directive, EPCR had no choice but to temporarily suspend the pool stage of the Heineken Champions Cup and the preliminary stage of the Challenge Cup."
It had already been confirmed that the 2021 Women's Six Nations has been postponed, with tournament organisers forced to concede defeat in attempts to stage the competition in its traditional February and March window. April and May have now been pencilled in as an alternative.
Exeter director of rugby Rob Baxter and Bristol Bears counterpart Pat Lam had stated their belief that Premiership games should be moved forward to free up fixture schedule space with games now likely to pile up for English clubs.
That would potentially create room for the pool matches in both European tournaments to be completed later this season on two current Premiership weekends.
However, shortly after the announcement that the Champions Cup and Challenge Cup were to be suspended, Premiership Rugby revealed they would be having a fortnight's hiatus.
Premiership Rugby chief executive, Darren Childs said: "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.
"The Gallagher Premiership Rugby season will return - as scheduled - on Friday 29 January when Bristol Bears host Bath Rugby."