The Argentina international was taken off at half time in Saturday's 2-1 Premier League win over Everton, although Postecoglou only described his issue as "tightness" after the contest.
As such, Romero was expected to make a full recovery in time for Thursday's trip to Brighton & Hove Albion, where Tottenham will already be without summer signing Micky van de Ven due to his own hamstring issue.
However, speaking to the media in Wednesday's pre-game press conference, Postecoglou announced that Romero's problem had proven more serious than first feared, and the 25-year-old will spend at least a month on the sidelines.
"Not so great with Romero, he had a scan the other day and he's got a hamstring strain so we're probably looking at four weeks or five weeks with him," Postecoglou told reporters.
As well as sitting out Thursday's clash with Brighton at the Amex, Romero's problem is also set to rule him out of imminent fixtures with Bournemouth, Burnley, Manchester United and Brentford before the end of January.
It also remains to be seen if the former Juventus man will shake off his affliction in time for Spurs' corresponding meeting with Everton at Goodison Park on January 3, and his absence leaves Eric Dier as Tottenham's only senior central defender at this juncture.
Postecoglou will likely call upon Dier - who will enter the final six months of his deal in January - to partner makeshift centre-back Ben Davies against the Seagulls, although 18-year-old Ashley Phillips could also be deployed in an emergency.
Postecoglou had already affirmed before Romero's injury that his side must try to bolster their backline in the January transfer window, with Nice's Jean-Clair Todibo and Genoa's Radu Dragusin thought to be shortlisted, but a move for the former may be difficult to pull off.
Romero is welcomed into the infirmary by Van de Ven, who established a solid partnership with the South American in the first two months of the campaign before picking up a severe hamstring concern of his own in November's 4-1 loss to Chelsea.
Last Friday, Van de Ven told The Sun at the World Darts Championships that he would be back in training in a couple of weeks' time, but Postecoglou has warned fans to take his words with a pinch of salt.
"I wouldn't put too much in what the players say mate. I don't think [he'll be back for] Burnley, maybe for Man United but I wouldn't know until I see him in training. He's at the tail end of it now," Postecoglou said of Van de Ven.
The centre-back duo are joined in the medical bay by ankle victims James Maddison, Rodrigo Bentancur and Alfie Whiteman, while Ivan Perisic and Manor Solomon are both on the road to recovery from knee operations, and Ryan Sessegnon is still nursing a thigh issue.
Richarlison also reported a sore back during the weekend's success over Everton, but Postecoglou has hinted that the Brazilian attacker will be fine to take part against Brighton.
"It was a pretty physically demanding game in terms of the numbers we generated against Everton, they were quite extraordinary. Fair to say the boys were a bit fatigued, but they've had a couple of easier days so everyone else should be OK," the Australian said.
Destiny Udogie is also available for selection again following a one-game suspension, but Yves Bissouma is banned for three more matches on account of picking up his second red card of the new term against Nottingham Forest.