It comes after the French government decreed its clubs should not play in the competitions this month.
As a result, European Professional Club Rugby (EPCR) says it had "no choice" but to postpone all matches scheduled for the weekends of 16 and 23 January.
However, EPCR aims to resume and finish the tournaments "as soon as practicable".
Tournament chiefs hope two rounds of Heineken Champions Cup and Challenge Cup pool games on the weekends of 15-17 and 22-24 can go ahead at a later date as authorities deal with the impact of a new strain of Covid-19.
