The 32-year-old had previously expressed his interest in following former team-mate Fredi Kanoute to China, but he is now poised to move to Turkey
The Super Lig outfit announced the news to KAP (Public Disclosure Platform) on Monday afternoon, just hours after coach Samet Aybaba admitted his interest in the France international.
Escude has been with Sevilla for six years but he has been actively seeking a transfer this summer, confessing last week that he would be interested in a move to China.
The 32-year-old centre-back, who arrived at the Ramon Sanchez Pizjuan from Ajax, has 13 France caps to his name.
Besiktas had also managed to persaude versatile Czech Republic international Tomas Sivok to ignore the alleged interest of Sevilla and sign a new deal with the club.