Dan du Preez, Ox Nche, Bongi Mbonambi, Scarra Ntubeni and Makazole Mapimpi, plus three of the management team, have also tested positive for coronavirus.
It takes the total number of cases to more than 20 in the Springbok camp.
However, a "large group" of South Africa players will return to training on Sunday after six days of isolation.
The three-Test series against the touring Lions takes place on consecutive weekends from 24 July.
Springbok coach Jacques Nienaber will remain in self-isolation after testing positive for Covid-19 on Monday, so director of rugby Rassie Erasmus will lead training.
"We are delighted to be able to return to the field and resume our preparations for the series," said Nienaber.
"It has been a challenging week with the Test against Georgia cancelled and the entire squad self-isolating for a few days as a preventative measure, so the players cannot wait to get back on the park."
South Africa's second Test against Georgia on 9 July was cancelled because both teams had Covid-19 cases.
"It is also great to have Rassie steering the ship while I am away," said Nienaber.
"He has been part of the system for several years now and he guided most of these players to the Rugby World Cup title in 2019, so the team is in great hands."