The South Korea skipper picked up a thigh injury against Qarabag in the Europa League and has missed Spurs' last two matches against Manchester United and Ferencvaros.
Son's issue was initially not believed to be too serious, but the 32-year-old did not make the trip to Hungary for Thursday's 2-1 Europa League win over Ferencvaros, being left in London alongside fellow injury victim Destiny Udogie.
The former Bayer Leverkusen man was initially called up to South Korea's squad for their upcoming World Cup qualifiers against Jordan and Iraq, but it was confirmed earlier this week that he had withdrawn from the national team ranks.
Straight after flying home from Thursday's victory, Spurs boss Postecoglou faced the media on Friday morning, where he confirmed that Son is "unlikely" to make the cut for Sunday's game.
Postecoglou positive on Destiny Udogie injury as Son blow confirmed
However, there is better news on left-back Destiny Udogie, who sat out the midweek success with a thigh injury sustained against Manchester United but is on track to be fit for the Seagulls showdown.
Postecoglou also confirmed that his side suffered no fresh blows on Thursday, telling the media: "First thing's first, we're all good after last night, tired, but physically, no issues.
"Of the guys who stayed back, Destiny, we think will be okay. We train today and tomorrow, so he has to get through that. Sonny is unlikely, he's pushing hard, but the turn around might be too quick for him to be available at this stage."
Son is likely to be one of three Tottenham absentees for the trip to the Amex, where Richarlison (unspecified) and Wilson Odobert (thigh) will also be missing, but Radu Dragusin is back from a European ban.
Could Tottenham start 17-year-old in place of Son?
With Son absent against Ferencvaros and the in-form duo of Dominic Solanke and Brennan Johnson rested from the first whistle, a youthful attack in Hungary featured teenage starlets Will Lankshear and Mikey Moore alongside Timo Werner.
Solanke will certainly displace Lankshear at the tip of the attack, while Johnson is also guaranteed to return on the right-hand side, but a fierce battle to deputise for Son on the left-hand side could materialise.
RB Leipzig loanee Werner was given the nod against Man United, but the German was guilty of wastefulness, firing two shots straight at Andre Onana with the rest of the goal at his mercy.
Another golden chance for Werner went begging against Ferencvaros, one set up by 17-year-old sensation Moore, whose performance on his first start for the Lilywhites earned him plenty of plaudits.
Postecoglou could not stop signing Moore's praises at the end of the evening, and the 2007-born starlet is now a genuine contender to start at the Amex ahead of the profligate Werner, although the latter's experience will work in his favour.