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.
Coach Rassie Erasmus Rassie Erasmus says the Springboks are fortunate to have good depth at lock within the squad and there is no need to replace Kleyn at this stage.
The reigning World Cup champions play Italy at Loftus Versfeld on Saturday in the first of two test matches, followed by a one-off test against Georgia at the Mbombela Stadium on July 19.
--ChannelAfrica--