The 26-year-old had only recently returned from a three-month injury layoff when he sustained the fresh blow in Sunday's 1-1 draw with Manchester United at Old Trafford.
Sakho opened the scoring for the Hammers in the second minute, but he was forced off the pitch 20 minutes into the second half after suffering the injury.
A statement on the club's official website read: "A scan on Tuesday revealed that the injury will keep him out of contention for up to six weeks. Everyone at West Ham United wishes him a speedy recovery and quick return to action."
Meanwhile, West Ham will face United at Old Trafford for the second time in four days as they prepare for tonight's EFL Cup quarter-final.