The Heineken Champions Cup is due to kick off on December 10, with Cardiff hosting European champions Toulouse the following day and Munster visiting Wasps 24 hours later.
Scarlets players and staff, meanwhile, are now understood to be isolating at a Belfast hotel after arriving into Dublin from South Africa during the early hours of Monday.
The Scarlets are scheduled to begin their European campaign against Bristol at Ashton Gate on December 11.
It is understood that there are no current plans for any of those European fixtures to be postponed.
Munster and Cardiff had hoped to join the Scarlets in leaving Cape Town on Sunday aboard a charter flight after South Africa was added to the UK Government's travel red list.
But Munster then reported one case of Covid-19, while Cardiff also remained in Cape Town following two positive cases, one of which is suspected to be the new variant Omicron, with both groups having started a period of isolation.
The United Rugby Championship had been scheduled to stage its first fixtures on South African soil over the weekend, but all four games were postponed.
"We are monitoring the situation and are in contact with the clubs," a European Professional Club Rugby spokesperson said.
In their latest statement, Cardiff said: "Cardiff Rugby continue to work with the South African Rugby Union, the Welsh Rugby Union and public health authorities both at home and abroad to decide the next course of action.
"The club are also continuing to work with all relevant authorities to secure the travelling party's return to Wales when safe and appropriate."
Round seven URC fixtures, scheduled to take place in South Africa next weekend, have also been postponed.
The outbreak of Omicron in South Africa has not yet affected Edinburgh Rugby and Glasgow Warriors, who have already welcomed South African teams to Scotland and were not due to play the reverse fixtures until spring.
The URC could, however, face a fixture headache if the situation worsens with an already congested calendar.