NSW police stopped Papley on Monday for what the club described as a roadside licence check.
"During the roadside stop, Papley completed a random drug test which returned a positive preliminary test indication," the Swans said in a statement on Wednesday.
"A second sample was taken for analysis.
"The results from the second test are expected to take at least four weeks."
Papley was behind the wheel, despite his drivers licence being suspended.
"Papley was issued with a court attendance notice for driving with a suspended licence due to non-payment of a parking fine," the Swans said.
The 30-year-old, renowned as one of the AFL's best small forwards, immediately told the Swans after being stopped by police, the club said.
Sydney, who then informed the AFL, said they would not comment further until the end of the police process.
Should Papley's second sample confirm a positive drugs finding, the 216-gamer is likely to face an AFL penalty separate from any legal sanction.
The Papley incident is the latest off-field drama at the Swans, who suspended five players for breaching club standards last August.
Vice-captain Isaac Heeney, Chad Warner, Nick Blakey, James Jordon and Riley Bice were stood down for the remainder of the AFL season and missed the finals.
The five were sanctioned for what the club described as a "serious breach of expected club behaviour and standards" after incidents at the Pullman Hotel in Melbourne following a round 23 game at the MCG.
The AFL-endorsed club bans were separate from a police investigation into an alleged sexual assault at the hotel, which is ongoing. No charges have been laid.
Bice is no longer a person of interest for police, while Blakey departed the Swans during last month's trade period to join North Melbourne.
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