Son missed Thursday's 1-0 victory at Eintracht Frankfurt with a foot injury, the 2-1 aggregate triumph taking Spurs through to only their second semi-final in a European competition in the last 40 years.
Postecoglou confirmed Son will be absent again for Nottingham Forest's Premier League visit on Monday, which comes only 10 days before the first leg of their Europa League showdown against Norwegian outfit Bodo/Glimt.
"Sonny's been dealing with his foot problem for a while. It's probably as good as time as any to give him the right time to get over it," he said.
Asked how long Son would be sidelined, Postecoglou replied: "I'm not really sure. We got to the point where he wasn't improving but it was deteriorating.
"I think probably rest is going to be as important as anything else with that kind of injury.
"We'll give him that time because Sonny is the kind of guy who always wants to train. We'll have to put him on ice for the next few days and see how he responds to that."
Son stayed in London on Thursday as Dominic Solanke's penalty, converted late in the first half, booked Tottenham's place in the last four of the Europa League.
The spot-kick came at a price as James Maddison was clattered by Frankfurt goalkeeper Kaua Santos and, according to Postecoglou, the England international is a "bit sore" and a doubt to play against Forest.
Beating Frankfurt kept Tottenham on course to win a first trophy in 17 years, since League Cup success in 2008.
But Spurs suffered a 17th Premier League defeat of the season in losing 2-0 at Wolves last weekend.
"It's fair to say it's been challenging. But part of that challenge has obviously been the rotations in the team," Postecoglou said of the Premier League campaign.
"It's going to be similar on Monday, we're going to have to make five or six changes to the team because we want to make sure everyone is in the right space."
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