The Australian has revealed the knee problem suffered in the first half of the 3-1 loss to RB Leipzig in the Champions League on Wednesday is “fairly significant” and it won’t be a "short-term" absence.
McGregor will see a specialist on Friday and will miss Saturday’s Premiership fixture with St Johnstone.
Celtic will get a better idea of the extent of the injury but he is likely to be missing for a sustained period of time.
"He's off to see a specialist today, it's fairly significant," Postecoglou told Sky Sports News.
McGregor was replaced by summer signing Oliver Abildgaard but the Celtic boss admitted that the team's captain is irreplaceable.
The 29-year-old has made over 30 appearances in each of the last eight seasons. Since the 2016/17 campaign he has missed just 17 games, playing 325 times.
"He's a fantastic player and beyond that, it's his influence on and off the field as a leader,” Postecoglou said.
"There's not one person that can replace that, it will be a case of the other guys stepping up and filling the breach.
"Obviously he is disappointed because he's such a big part of how we play our football and what we've created here so I know he'll be working hard, whatever the timeline to get back."
Celtic have had a few injury issues recently with Carl Starfelt and Cameron Carter-Vickers absent, as well as David Turnbull.
They have another seven games in October alone and 11 before the league breaks for the World Cup in November.