The 26-year-old hurt his left ankle during the first half of Bath's 41-6 defeat to Saracens on Sunday and needed crutches to leave the field before his foot was put into a protective boot.
"JJ is going to need surgery to correct his injury. He will then need a lengthy period of time to rehabilitate," Bath's Director of Rugby Todd Blackadder told the club website.
"Our medical team will work closely with him to ensure we get him back on the pitch as soon as possible."
England start the tour against the Springboks in Johannesburg on June 9 and Joseph, capped 40 times, had been due to play a prominent role.
(Reporting by Alan Baldwin, editing by Toby Davis)
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