Argentina international Romero came off at half-time during their 2-1 weekend home victory over Everton. As a result, Spurs are now without their two best central defenders with Dutchman Micky van de Ven sidelined since early November and not expected to return until next month.
"No, no. He's no good. He's had a scan, so he'll be out for four or five weeks," Postecoglou said of Romero's absence ahead of Thursday's Premier League trip to Brighton.
"Everyone else got through it okay. There's a few sore bodies there but we've had a, sort of, easier training week, last few days anyway, but everyone else is okay."
James Maddison, Ivan Perisic, Ryan Sessegnon, Manor Solomon, Rodrigo Bentancur and Ashley Phillips are the other long-term absentees in the Tottenham squad.
Postecoglou said Van de Ven could return for their trip to Manchester United on January 14 and acknowledged they need another centre-back to provide some backup in that position.
"I don't put as much stock into the left-sided or right-sided centre-back scenario, we just need another player in that position who can provide some more depth for us and the ability to change it up," he said.
"It is more the profile and characteristics. We didn't go looking for a left-sided centre-back at the start of the year, it just happened that Micky was the perfect centre-back.
"It is about looking at the right profile, the right characteristics and seeing what is available, then going from there."
Postecoglou was also full of admiration for Brighton and how the south-coast club is managed by Roberto De Zerbi.
Brighton are ninth in the league and through to the Europa League knockout stage in their first season in Europe.
"Their football is fantastic, they do things a little differently to everyone else," Postecoglou said.
"They're a very well run club, they have been for quite a while ... Roberto's come in and he's always done things fairly unique in his career and he's brought that to the Premier League. You've got to respect that.
"This is probably the toughest league in the world and it's easy to shy away from doing anything that's too different. But you can tell he's determined to do things his own way."
