"He's got a fractured cheekbone and eye-socket," manager David Moyes, speaking ahead of Wednesday's FA Cup fifth-round tie at Manchester United, told a news conference.
"We don't quite know whether it'll be operated on or healing itself, or a time-span at the moment."
Fabianski's injury is a blow to West Ham, who are two points above the relegation zone with 14 league games left.
The 37-year-old has started every game in the Premier League this season.
West Ham will now have to turn to second-choice keeper Alphonse Areola, who came on against Forest after Fabianski was injured.
"We've had the surgeons report back but I think there's a decision to make as to whether it's surgery. It's huge for us and huge for the goalkeeper as well.
"The stage of the career for him as well."
Moyes confirmed that West Ham will also be without right back Vladimir Coufal for the trip to Old Trafford because of a heel injury.
Striker Danny Ings, who scored his first two goals for the Hammers against Forest, is also not available as he is cup-tied after appearing previously in the competition for Aston Villa.
