Mark Alexander, president of SA Rugby, said the eight-match tour schedule was subject to review given the constantly changing nature of pandemic. The games could be played without fans.
"We appreciate the Lions' faith and share their desire to see a safe and successful tour," Alexander said.
"We have been in regular contact with our government to make that a reality against the backdrop of the pandemic and its predicted progression over the coming months," he added. "There are serious financial implications for SA Rugby, should the event take place without any supporters in attendance, and we cannot ignore that in our considerations."
Bill Sweeney, chief executive of England's Rugby Football Union, effectively ruled out the option of staging matches in Britain and Ireland, saying "we're just running out of runway."
An option of Australia hosting the tour also "has gone away," Sweeney said.
"If you look at the original agreement, it says South Africa have an obligation to host the tour in South Africa and the Lions have an obligation to turn up and play," said Sweeney, speaking before the joint announcement by the teams. "No one has ever said that can't be fulfilled. What we're now saying is can we have absolute clarity on that, can you still host the Lions tour? We believe they're saying they can."
Sweeney said a Lions tour going ahead without fans would be "quite controversial," adding: "It's not in the spirit of the Lions and we're very conscious of that."
Warren Gatland's side are scheduled to play eight games on the tour, with the final three being the test match series.
The Lions last toured South Africa in 2009 with the hosts, who were defending World Cup winners as they are 12 years later, winning the series 2-1.
