Kleyn, who started the match with lock partner Lood de Jager, will return to Irish club Munster to recover from the injury and undergo the necessary rehabilitation.
The Springboks made the journey from Cape Town to Johannesburg on Sunday to begin their preparations for the opening match of the Castle Lager Incoming Series against Italy at Loftus Versfeld in Pretoria on Saturday, 5 July.
They will line up against Italy again a week later at the Nelson Mandela Bay stadium in Gqeberha before taking on Georgia at the Mbombela Stadium in Nelspruit in the closing match of the series on Saturday 19 July.
“We feel for Jean,” said Springbok coach Rassie Erasmus, “but injuries are part of the game, and we are fortunate to have good depth at lock within the squad, so there is no need for us to replace him at this stage.”
The team had Sunday off to travel and recover from Saturday’s encounter and will begin their on-field preparations for the first official Test match of the season on Monday.