The forward was in a rich vein of form recently, scoring 11 goals including 10 in the Premier League, but picked up the injury in a 2-1 home loss to Wolverhampton Wanderers last month.
"Richarlison is out for three to four weeks. He picked up a bit of a knee injury in the last game," Postecoglou told reporters ahead of Saturday's game with visiting Crystal Palace.
"It's disappointing as he's been a key contributor but that's how our season has gone. We get back, we lose one, we'll have to compensate him being out."
Postecoglou said Ryan Sessegnon has also had a setback after an operation on his right hamstring this week. The wing-back has had several muscle injuries in recent seasons.
"It's a tough one, it's through no fault of his own, he's worked hard to come back," Postecoglou said.
"It's always challenging when it happens as a series of events and he hasn't had any reward for all his hard work of rehab in terms of getting out there to play.
"He's had surgery, he's a positive guy and he's been through this before, so hopefully that helps."
Postecoglou said left-back Destiny Udogie could return from a knee injury after training well but right-back Pedro Porro needs another week to recover from a strain.
Spurs are fifth in the table, five points behind fourth-placed Aston Villa with a game in hand, and Postecoglou said the two-week break after losing at home to Wolves was not ideal.
Spurs did not play last weekend with their scheduled opponents Chelsea involved in the League Cup final.
"It's been a very disruptive season for us, apart from the first 10 or 11 games. We had momentum and rhythm," added Postecoglou.
"We've had stops and starts to our season. We've got a couple of games then there's an international break. At least after that, we'll have regular football until the end of the year."
