The Argentina international sustained the issue during Spurs' 2-1 beating of Everton on December 23, and Postecoglou admitted at the time that he would likely be out for four to five weeks.
As a result, Romero has been forced to miss Tottenham's last three games in all competitions, and the initial expectation was that he was unlikely to represent the club again until the end of January or beginning of February.
However, reports earlier this week claimed that the World Cup winner was back in training with the North London giants ahead of schedule and would be in contention to make his return to first-team action at the Theatre of Dreams.
Speaking to the media in his pre-game press conference, Postecoglou confirmed that Romero had begun working with the group again, but there is only a slim chance that he will be involved against the Red Devils.
"The only one to join the main group today is Romero and that's it I think. He's just started training so it's fair to say it might be a bit close this weekend," football.london quotes Postecoglou as saying.
With Romero unlikely to be involved from the first whistle, Postecoglou could hand an immediate debut to new signing Radu Dragusin, whose arrival from Genoa on a long-term contract was confirmed on Thursday evening.
Dragusin has been registered as a Tottenham player in time to play at Old Trafford, and Postecoglou is hopeful that the Romanian's work permit will be sorted out in time for him to be part of the matchday squad this weekend.
"From what I'm told it shouldn't be a problem. If all the relevant paperwork comes through he'll be in the squad," the Australian added. "We knew that was a priority position for us in the window. We knew that if the right opportunity came up in January we'd move.
"We did a fair bit of work on it. Once we knew the direction we were going in, Johan [Lange] and his team did a lot of background work on him. After my chats with him I could tell how he would fit in."
Dragusin could form a new-look central pairing with Micky van de Ven, who was back on the bench against Burnley after recovering from his serious hamstring issue, while Ryan Sessegnon also made his first appearance of the season against the Clarets following thigh surgery.
However, Ben Davies and Giovani Lo Celso both picked up injuries against Burnley, which will rule them out of Sunday's game alongside Ivan Perisic (knee), Manor Solomon (knee), Alejo Veliz (knee), James Maddison (ankle) and Dane Scarlett (muscle).
"From last week, Ben Davies came off with a hamstring - probably looking at a month or so with him. Gio Lo Celso, similar, but it's a bit harder to diagnose where he's at so we'll send him for tests on Monday but he's out. Dane Scarlett has picked up an injury in training so he's out. Both muscle injuries - Dane hamstring, Gio muscle," Postecoglou added.
Pape Sarr, Yves Bissouma and Son Heung-min are also away on international duty, but Timo Werner could be in line for his second Premier League debut against Man United, having snubbed the Red Devils to join Spurs on loan from RB Leipzig.
Tottenham return to top-flight action sat fifth in the Premier League table, but with Arsenal not involved in the depleted gameweek, the Lilywhites will rise into the Champions League spots with victory at Old Trafford.