Contingency plans had been drawn up, which included bringing the matches to Britain and Ireland or postponing the tour until 2022 or 2025, because of the coronavirus pandemic.
But the two teams said they were “aligned” on delivering the tour “in the scheduled playing window.”
“I can confirm that the board's intended position is for the tour to go ahead as scheduled in South Africa in 2021,” said Jason Leonard, chairman of The British & Irish Lions.
“We acknowledge that there is a significant amount of work still to be undertaken to deliver a robust Covid-19 countermeasure plan to ensure a successful, safe and uninterrupted tour. SA Rugby will have our full support to help implement this plan.”
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.”