The Spurs captain was rested for Thursday's 1-0 win over AZ Alkmaar in the Europa League after struggling with soreness in the wake of last weekend's 4-1 thrashing of West Ham United.
Postecoglou revealed before the midweek success that it was always the plan to give Son a rest in midweek, and the South Korean was initially thought to have been in with a good chance of being available against Palace at Selhurst Park.
However, speaking to the media in his pre-game press conference, Postecoglou revealed that his skipper was not set to take part in training on Friday and is therefore unlikely to make the cut for the derby.
"Sonny's still not right, still not feeling 100 per cent. He won't train today, so he's unlikely for this weekend and we'll see after that," Postecoglou said, before going on to confirm that his side came through Thursday's win with no new worries.
Tottenham likely to be missing two players against Palace
"In terms of last night (1-0 win over AZ Alkmaar), we're all good, no issues, a bit of fatigue for some of the guys where they've not played for a while," the Australian added.
While there is still the slimmest chance that Son will be available for the weekend, the 32-year-old is expected to take his seat on the sidelines as one of two Spurs players missing out through injury.
Right-back Djed Spence is also out of contention with a groin problem, and Postecoglou confirmed that the defender should be back in full training next week, potentially pointing to a comeback against Manchester City in the EFL Cup or Aston Villa in the Premier League next weekend.
Spurs enter the weekend occupying seventh place in the Premier League table after eight matches, and they could end the weekend as high as fifth should they win and both Brighton & Hove Albion and Chelsea suffer defeat.
Could a 17-year-old replace Son in Palace clash?
As influential as Son is to Tottenham, Postecoglou is not short of alternatives to the ex-Bayer Leverkusen man in the final third, and as many as four players could be battling to fill in for him this weekend.
One of those is 17-year-old sensation Mikey Moore, who had already sent tongues wagging before Thursday's visit of AZ Alkmaar and earned even more plaudits with a dazzling display against the Dutch outfit.
Moore had three shots and created one chance in the 1-0 victory, during which he operated on both sides after Timo Werner - whose wastefulness is showing no signs of ending anytime soon - was taken off at half time.
Werner might be the most obvious choice to replace Son, but the German is not cutting a confident figure in front of goal, and Postecoglou also has the option of using Wilson Odobert following his recovery from a thigh problem.
Alternatively, Richarlison could operate out wide after netting the only goal of the midweek victory from the penalty spot, although the Brazilian is expected to drop down to the bench for this one.