Farrell limped off during England's 32-15 victory over Australia at Twickenham last Sunday (AEDT).
Saracens on Friday revealed the England skipper will be out for 10 to 12 weeks, with an operation required.
The versatile Farrell faces a battle to be fit for the Red Rose's opening game of the 2022 Six Nations against Scotland at Murrayfield on 6 February.
Sarries also confirmed in a fitness bulletin that their England hooker Jamie George will be sidelined for eight to 10 weeks due to the knee damage he sustained in the win over the Wallabies.
The Premiership club was able to offer positive news of Elliot Daly, who is back in full training and available to return after recovering from a leg injury.
England head coach Eddie Jones has named Courtney Lawes as captain for the clash with South Africa on Sunday in the absence of Farrell.
Manu Tuilagi moves to inside centre to fill the void left by Farrell, with Joe Marchant starting on the right wing.
Rookie duo Bevan Rodd and Jamie Blamire will start in the front row, while hooker Nic Dolly could make his debut off the bench against the world champions.