It's a development that could also have knock-on consequences for the Gunners' interest in Christos Tzolis.
Trossard, 31, had appeared close to a switch to the Turkish side this summer, with the Super Lig club believed to be prepared to pay in the region of €18m and the player's personal terms understood to have already been settled. Completion of any deal had been pencilled in for after the World Cup, but it now appears that isn't as close as first indicated.
Agent Dirk Hebel told Sky in Germany that Trossard is continuing to weigh up his options.
"I can confirm that Besiktas did everything they could to sign Leandro Trossard," he said. "However, no decision has been made yet.
"Leandro has all the offers on the table and will make his decision soon. He is a world-class player with several options, including staying at Arsenal."
Should Trossard opt to stay in north London, it could complicate Arsenal's efforts to sign Club Brugge forward Tzolis, who has emerged as an ideal like-for-like replacement for the Belgian on the left flank. A fee in the region of £35m is understood to have already been agreed in principle for the Greece international, but Arsenal's need for the Brugge man would diminish significantly if Trossard commits his future to the club.
Arsenal will be watching Trossard's decision closely over the coming days, with the outcome likely to shape how the club approaches the rest of their business on the left side of attack before the new season begins.
Trossard has been a regular fixture in Mikel Arteta's side since arriving from Brighton, and losing him on a free run-down of his contract in 2027 without any fee would represent a significant blow given his versatility across the front line.
A cut-price exit to Besiktas this summer would at least allow Arsenal to recoup some value while freeing up space and budget to pursue Tzolis or other attacking targets they have been monitoring.