It is the latest challenge for Smith as he struggles for form and Wednesday's news comes a day after coach Luke Beveridge strongly defended the midfielder.
Beveridge said Smith was unwell for last week's loss to Collingwood, when he only managed 11 possessions.
"The 22-year-old presented to the club yesterday for a main session, but wasn't well enough to participate and subsequently won't make the trip to Sydney today," the Bulldogs said in a statement.
One of the AFL's most marketable players, Smith's form has become a major talking point in the media and there has been discussion about whether he should move clubs.
That prompted a strong response from Beveridge, who labelled the commentary "pungent" and said there were reasons why the midfielder had been below his explosive best.
"All these things that people aren't aware of, and if you haven't got the information, you don't know the facts, then all you do is do harm," he said.
Smith's profile has skyrocketed since being drafted by the Bulldogs in 2018, becoming one of Australian sport's most recognisable faces.
He instantly had an impact for the Dogs on the field, playing a starring role in the club's extraordinary charge to the 2021 grand final.
Smith has a huge social media following and has a strong connection to younger fans for openly discussing his mental health challenges.
He received a two-game suspension by the AFL last year for conduct unbecoming after photographs and video emerged of him using illicit drugs.
"We all see the potential of a young man who should be a really influential player in the competition," Beveridge said.
"We're hoping that, whether it's this year, or into next year, that he finds his way to his absolute best and we're just going to help him and we believe he can get there."
Meanwhile, Beveridge said defender Ryan Gardner remained a chance to face the Swans after hurting his ankle against Collingwood.
The Bulldogs are seventh, while Sydney are 15th and two games outside the eight, badly needing a win to stay in the finals hunt.
The Swans held a morning tea for coach John Longmire on Wednesday, the coach bashfully cutting a cake to celebrate his 300th game in charge.
"No he's not (one for milestones)," midfielder James Rowbottom said of the coach.
"It was nice to touch on; Horse has been a massive part of the club for the last two decades or so, it's a massive achievement.
"To see how Horse has evolved and changed in my time here has been amazing; he's a real second-father figure."
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