Whiteley sustained the injury in the first half of Saturday’s 19-17 Super Rugby loss to the Stormers, with his diagnosis confirmed by the team on Monday.
The 31-year-old will miss a sizeable portion of the Super Rugby season, but will still have time to confirm fitness and form ahead of the World Cup in Japan, where he is likely to be understudy to Duane Vermeulen in the number eight jersey.
Whiteley has endured an injury-ravaged past two years and missed most of the 2018 Super Rugby season with a knee problem.
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The Lions are next in action when they host the Bulls at Ellis Park on Saturday.
(Reporting By Nick Said, editing by Pritha Sarkar)
