Gestede has scored two goals in Villa's first three games of the Championship season, prompting Besiktas to prepare a serious offer for the Benin international.
Two Bundesliga clubs are also believed to be interested in the 26-year-old who signed a five-year contract at Villa last summer after joining for around £6m from Blackburn Rovers.
Gestede is believed to be happy at Villa, whose ambitious new Chinese owner Tony Xia is targeting an immediate return to the Premier League.
But Besiktas are preparing to put an offer on the table they believe could tempt both club and player. President Fikret Orman is on the hunt for a new striker after losing German international Mario Gomez earlier this summer.
Gomez scored 26 times as the Istanbul club won the Turkish Super Lig last season but opted not to extend his loan stay, citing the "political situation" in the country following a failed military coup last month.
Chelsea's Loic Remy and Manchester City's Wilfried Bony have also been linked with a possible move to Besiktas, who have been beaten to Christian Benteke's signature by Crystal Palace.