Burger has not played since he tore medial ligaments while captaining the Stormers on the opening weekend of the Super 15 season in February.
The 29-year-old's had hoped to return for the opening Test on June 9 but, after missing the Stormers' recently-completed four-week tour, the club admitted on Friday that they did not expect him to recover in time.
"Schalk Burger, Duane Vermeulen and Nick Koster (all knee injuries) are expected to return only after the June break," a statement on the club's website read.
Vermeulen's exclusion also comes as a blow to the Springboks after the number eight enjoyed a good start to the season before also succumbing to a knee injury.
Burger was at the centre of controversy three years ago when he was banned for eight weeks after he was found guilty of eye gouging British and Irish Lions winger Luke Fitzgerald.