According to Turkish outlet Fotomac, Galatasaray have moved to position themselves for a move for the Sweden international, who has become a peripheral figure under Mikel Arteta this season.
Gyökeres arrived at Arsenal from Sporting CP last year for a reported fee in the region of £57 million, after scoring 21 goals and providing three assists in 55 matches during his debut campaign in England.
This season, though, the 28-year-old has featured just twice from the bench in five Premier League matches, with Arteta preferring alternative options in attack.
Turkish reports claim Galatasaray have acted decisively as they eye the January window, describing the forward as unsettled and eager for regular football elsewhere.
Interest is not confined to Turkey. Italian giants Milan and Juventus have also been credited with checking on Gyökeres's availability, while Saudi clubs Al Ahli and Al-Ittihad have likewise been linked with a move, reflecting the scale of frustration surrounding his lack of minutes.
Should Galatasaray press ahead, they could offer Gyökeres regular continental football alongside attacking talents such as Victor Osimhen, Leroy Sané and Rafael Leão.
Gyökeres has scored 21 goals in 37 caps for Sweden and remains prolific on the international stage even as his club form has stalled, having previously fired Sporting to back-to-back league titles before his move to England.
He remains under contract with Arsenal until 2030, meaning any interested party would need to negotiate a significant fee even if the player himself pushes for a January exit.
Before his move to England, Gyökeres spent two seasons in Portugal, where he built a reputation as one of Europe's most prolific strikers, netting close to 100 goals for Sporting and winning the Primeira Liga in successive campaigns. That form persuaded Arsenal to move for him after a lengthy chase, but his early struggles for game time in north London stand in sharp contrast to the numbers he produced before arriving.
Arsenal have not commented publicly on the striker's situation, and Arteta has continued to rotate his forward options between Gyökeres and alternatives in recent weeks. With the January window still some months away, how Gyökeres responds between now and then could yet determine whether suitors such as Galatasaray follow through.