Cardiff and the Scarlets say they are struggling to fly their teams back from South Africa despite securing a charter plane.
The two Welsh regions, along with Munster and Zebre Parma, have been in the country preparing to play back-to-back matches in the United Rugby Championship.
Those fixtures have since been postponed after South Africa was added to the UK Government's red travel list.
A Cardiff statement read: "Cardiff Rugby continue to work tirelessly to repatriate their travelling party from South Africa following sudden developments around the Covid-19 variant B.1.1.529.
"A charter aircraft has been secured, however we have been unable to secure Civil Aviation Authority clearance due to the closure of borders in the UK and European Union."
A Scarlets statement added: "Together with our colleagues at Cardiff Rugby, we had hoped to fly out of South Africa on Friday."
The URC reacted to news of the discovery of a new coronavirus variant by postponing the next two rounds of matches on South African soil, starting with this weekend's games between the Stormers and Zebre, the Sharks and the Scarlets, the Bulls and Munster, and the Lions and Cardiff.
The URC said in a statement: "Due to the sudden developments connected with the new Covid-19 variant (B.1.1.529) that have immediately placed South Africa on the UK and EU travel red lists, the scheduled Round 6 and 7 United Rugby Championship fixtures set to take place in South Africa over the next two weekends have been postponed and will be rescheduled later this season.
"The safety and well-being of our participating clubs' players, coaches, support staff an match officials is the foremost priority and the URC is currently working with the four visiting clubs - Cardiff Rugby, Munster Rugby, Scarlets and Zebre Parma - to facilitate their return as soon as possible."
UK Health Secretary Sajid Javid said flights to England from South Africa, Namibia, Lesotho, Botswana, Eswatini and Zimbabwe will be suspended from midday Friday and all six countries will be added to the red list.
Anyone arriving after 4am on Saturday will need to stay at a managed quarantine hotel.
That would have a major impact on the two Welsh regions' preparations for their opening Champions Cup matches.
Transport Secretary Grant Shapps said on Friday the UK was "buying time" by adding South Africa, Botswana, Lesotho, Eswatini, Zimbabwe and Namibia to its travel red list, adding that the Government was taking a "safety-first approach" over the B.1.1.529 variant.
Golf's European Tour - now known as the DP World Tour - has just commenced a three-week swing of events in South Africa and there were numerous withdrawals from the Joburg Open.
Englishmen Oliver Fisher, Steven Brown, Richard McEvoy, Matthew Jordan, Andrew Wilson, Matt Ford and Graeme Storm, Scots Richie Ramsay, Liam Johnston and Craig Howie, Northern Ireland's Jonathan Caldwell and Cormac Sharvin and Welshman Oliver Farr all pulled out of the tournament before the second round commenced on Friday, along with Irish pair Niall Kearney and Paul Dunne.