The 22-year-old England Under-21 international has become the club's first-choice keeper this season, making 16 appearances in the Premier League.
"The good news is he hasn't ruptured his ACL (anterior cruciate ligament)," said boss David Moyes. "The bad news is he's strained it.
"It's a major disappointment because he's been saving us points."
Pickford suffered the injury day in the 3-1 defeat by Manchester United on Boxing Day, but finished the game.
Vito Mannone, who played the first two games this season before being ruled out for three months with an elbow injury, is likely to start on New Year's Eve at Burnley.
"Vito is fine and he's been training for many weeks since his injury. He has a lot of experience," said Moyes.
The Black Cats also have 25-year-old Mika, who is still waiting for a debut after joining from Boavista in September.
Sunderland are in the relegation zone, two points below Crystal Palace.
Sunderland goalkeeper Jordan Pickford has been ruled out for six to eight weeks with a knee injury.