The 32-year-old was a notable absentee as Tottenham progressed to the Europa League semi-finals with a 1-0 win over Eintracht Frankfurt on Thursday.
Son has now missed Spurs’ last two games having sat out their 4-2 Premier League loss to Wolves earlier in the month.
Speaking ahead of their game with high-flying Nottingham Forest, Postecoglu was less than optimistic about the forward’s situation.
"Not really sure. Like I said, it's something that he's been dealing with for a little while, and we've got to the point where it was not only not improving, it was deteriorating," he revlealed.
"It's one of these things, I think probably rest is going to be as important as anything else with that kind of injury. So we'll give him that time, because Sonny’s the kind of guy that always wants to train. We'll just have to put him on ice for the next few days and see how he responds to that."