Solomon has featured in five of Spurs' seven Premier League games so far this season, but his manager will now have to plan without him for their upcoming games, including Saturday's trip to Luton Town in the Premier League.
"Manor Solomon unfortunately during training got a significant injury to his knee, he'll miss some significant game time over the next few months," Postecoglou told reporters.
"That's disappointing because he was contributing for us already and growing into his role. With Ivan Perisic already down, it's not ideal when you lose a player in that position."
There was better news on the condition of Son Heung-min and James Maddison, who both came off early in the games with Liverpool and Arsenal to protect them from knocks they've been carrying.
"Everyone else from last week is okay. Sonny (Heung-min) is fine. He trained the last two days, no problem," the manager said.
"Maddison has worked hard all week, he hasn't missed a session," he added.
Spurs have made an unbeaten start to their league season, but the 2-1 win against Liverpool last weekend was overshadowed by controversy over an opposition goal that was wrongly disallowed after a VAR communication error.
Postecoglou was unaware of what had happened during the game, but said that even if he did have knowledge of the situation, he could not see himself making a decision to allow Liverpool to score to make up for the error.
"We've got pretty hefty responsibilities at our clubs but we're not the custodians. I wouldn't make a decision that could potentially send our club down, on the back of what my beliefs are," he said.
Spurs are second in the standings, level with Arsenal on 17 points and one behind leaders Manchester City. With those two clubs facing each other on Sunday, a win at Luton, in 17th place, on Saturday would put Spurs on top.
"Look, if we win we go top of the table for 24 hours but who cares? We've got to win," Postecoglou said.
"It's a tough game, our focus is on putting on a performance and playing at our best because if we don't it won't matter where we're sitting on the ladder, it'll be us focusing on a poor performance."
