The Spurs skipper returned from a hamstring problem in the weekend's 4-1 thrashing of West Ham United in the Premier League, marking his comeback with a goal against Julen Lopetegui's side.
However, Son was a notable absentee from Spurs' training session on Wednesday morning, fuelling fears that he may have suffered a fresh setback after spending three games out with his initial hamstring problem.
Postecoglou was asked about Son's fitness straight away in his pre-game press conference, and the Australian confirmed that the 32-year-old would not be involved on Thursday, although that was always the plan regardless of how he fared at the weekend.
Djed Spence was also absent from team training after picking up a groin strain last week, but Postecoglou did not deliver a detailed update on him, saying: "Sonny was a bit sore after the weekend, he was never going to be involved tomorrow anyway because he missed a bit," as quoted by football.london.
Postecoglou provides positive Richarlison, Wilson Odobert updates
"So we are just leaving him out of tomorrow and see how he is from there. Djed Spence trained individually today. Everyone else was out there."
Son's absence could open the door for Richarlison or Wilson Odobert to earn some minutes in his place, as the former endeavours to make his first start of the season in any competition after returning from his own issue off the bench at the weekend.
Asked if the Brazilian could start Thursday's match, Postecoglou replied: "Yeah, yeah. Richy got through the weekend really well, he's trained, this is almost the second week of training now, so he's ready to go in terms of being available, for sure."
However, the Tottenham boss is unsurprisingly taking a more cautious approach with Odobert, who has missed Spurs' last six games in all competitions with the hamstring injury he suffered against Coventry City in the EFL Cup last month.
"Yeah, potentially," Postecoglou replied when asked if Odobert could see the field against AZ. "Again, we've got to be a bit more careful with him because he's missed a lot.
"He's only had this week of training. He'll be involved tomorrow, in the squad. Game-time will be game-dependent with him but good to get him back and after this week he should be fully available."
Postecoglou responds to Tottenham fine from West Ham incident
Earlier on Wednesday, Tottenham were fined £20,000 by the Football Association after admitting to a charge of failing to control their players in the win over West Ham, where a late fracas saw Irons attacker Mohammed Kudus sent off.
Postecoglou was loath to discuss the incident in detail, though, saying: "We've been fined. All I said was I think we handled it the best we could but the FA thought it wasn't a good look and has fined us for it."
Tottenham boast a perfect record of six points from their first two games of the 2024-25 Europa League, seeing off Qarabag 3-1 before a nervy 2-1 win over Ferencvaros at the start of October.
Meanwhile, AZ Alkmaar triumphed 3-2 over Elfsborg in their opening battle before losing 2-0 to La Liga outfit Athletic Bilbao on matchday two.