The squad is training in Cape Town.
South Africa has been put on the UK red travel list.
The URC matches would be called off.
Six countries will be added to the red list on Friday, according to UK Health Secretary Sajid Javid.
Any British resident who arrives after 04:00 on Sunday must stay in a hotel.
Those returning before that would be asked to stay at home for 10 days.
Some African countries are on the red list with flights temporarily suspended.
The Welsh government has adopted travel rules for England in recent times, and any flight from South Africa to England would usually land in England.
Charter flights are being considered as the teams try to return home on Friday before the deadline.
The news about the new Covid variant that affects our people in South Africa will have been seen by many.
We are focused on the welfare of our staff in South Africa and their families at home. We are looking at all options to get our people home.
The two teams were going to play two games in South Africa. They were going to return to the UK for the European competition.
The URC has teams from Ireland, Italy, and South Africa.
The competition was supposed to be the first time that Welsh teams played in South Africa.
The games between the Lions and the Scarlets were to be played in South Africa on the same day.
Both Welsh sides were to remain in South Africa, with the Scarlets playing the Bulls on Friday, 3 December at Versfeld and the Stormers playing the Stormers the following evening.
The games are not expected to go ahead, but if the squad can return before the holiday, the games could be played.
If the players from the two Welsh teams have to stay at home for 10 days on their arrival in the UK, the European games would still be in danger.
The initial plan was for South African sides to return to Europe in November and December.
In October, competition bosses confirmed that the home games for the Sharks, Stormers, Lions and Bulls would be in round six and seven.
South Africa was removed from the UK government's travel red list, which meant that there was no need for scurvy on return.