The return of Argentina's Christian Romero from suspension has helped alleviate their problems but centre-backs Micky van der Ven, Ashley Phillips and Eric Dier are all injured.
Wales full-back Ben Davies has been filling in but Postecoglou says it is an area that need reinforcing.
"I thought Ben Davies was outstanding (in Sunday's win against Newcastle United). He's getting a run of games," Postecoglou, told reporters ahead of Friday's Premier League visit to lowly Nottingham Forest.
"We've got to be mindful he's not a centre-back and he's growing into that role. Ultimately we need to sign another centre-back though because at the moment we're on tenterhooks in case something else happens."
Tottenham snapped a run of five games without a victory in the 4-1 defeat of visiting Newcastle and are well positioned in fifth spot as they prepare to take on 16th-placed Forest.
Several players remain unavailable through injury including forwards James Maddison and Manor Solomon and midfielder Rodrigo Bentancur and there was another blow this week with Giovani Lo Celso ruled out of the trip to the Midlands on Friday.
"Unfortunately he got a bit of a niggle at training, I don't know what it is, but he missed training today so he's not available tomorrow. Hopefully, it's nothing too serious," Postecoglou said.
Victory would take Tottenham level on 33 points with fourth-placed champions Manchester City.
