Bentancur, who only recently returned from a nine-month lay-off because of a serious knee injury, hobbled off in the first half of Spurs' 2-1 Premier League defeat by Aston Villa last Sunday.
"He'll probably miss a fair chunk of time between now and the New Year," Postecoglou said ahead of Sunday's game at champions Manchester City.
"Probably a couple of months for him to get back, which is disappointing for him because we all know and have witnessed how hard he has worked to get himself back."
Bentancur was making his first start for the club since February when he got injured.
"He'd done everything right, he had a good international break where he got back playing for his country... with all that context you're just gutted for him, because of the person he is and what he's gone through," Postecoglou told reporters.
"He's still such a positive guy. I caught up with him yesterday and he's just again determined to crack on and get back as quickly as possible."
Tottenham have a long injury list, as well as suspensions, with key players such as England forward James Maddison and Dutch defender Micky van de Ven among them.
Postecoglou said Pape Matar Sarr was also unavailable with a hamstring problem but offered better news on Brazil forward Richarlison.
"Richy is probably the closest out of the ones that have been missing. We're looking at getting him involved in training hopefully tomorrow and then we'll see how he is because he has obviously missed a fair bit," he said.
"He's done his rehab really well and he feels good which is important. He's sort of the next cab off the ranks in terms of the injured guys."
Postecoglou said Spurs may have to be more active in the January transfer window as a consequence of the injuries.
"We may be forced to, we may have no choice on it," he said.
"We've got some sort of specific ideas about what we're going to try and do, for sure, but at the same time depending on how things go between now and then and how the recovery of guys is we may need to make a different kind of decision."
