Cleary, who has played five senior games, has also been handed a suspended $5000 for an adverse finding under the league's illicit drugs policy.
The penalties relate to an incident in Sydney's eastern suburbs in December last year, when police caught Cleary trying to buy drugs.
The 19-year-old received a criminal infringement notice for possession of an illicit substance and later received a $400 fine, with no criminal conviction recorded.
A first-round draft pick in 2023 from the Swans' academy, Cleary chose not to notify the club at the time.
"Cleary is extremely remorseful and has cooperated fully with the AFL investigation," Sydney said in a club statement.
"The club is extremely disappointed and does not condone nor tolerate illicit drug behaviour.
"The Sydney Swans will continue to educate our players on the pitfalls of such behaviour."
Cleary will sit out of Sydney's blockbuster fixtures against Hawthorn and the Brisbane Lions, and is also ineligible to feature in the Swans' community series fixture.
Hawks forward Jack Ginnivan, then with Collingwood, and Carlton's Elijah Hollands also served two-match drugs bans over the last two seasons.
Former Melbourne player Joel Smith, who returned a positive test for cocaine in 2023, is banned until 2028 because of five anti-doping rule violations under the Australian Football Anti-Doping Code.
"AFL Players are well-educated when it comes to the harms associated with the use of illicit substances, and have a responsibility to themselves, their clubs and their careers to uphold community expectations," AFL general counsel Stephen Meade said in a statement.
Cleary played five matches in his debut season last year, including three as the substitute, but did not feature in the Swans' finals campaign.
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