The Portugal international, 34, drew the curtain on his spell at the Santiago Bernabeu after 10 years of service in June, leaving on a free transfer.
And Besiktas have moved to snap up the out-of-contract centre-back, after announcing talks had begun between the two parties on their official website.
"Negotiations have begun with Kepler Laveran Lima Ferreira (Pepe) over the player's transfer," a short statement read.
The announcement of talks is a necessary formality due to Besiktas' status as a publicly traded company, which requires them to release details of all financial transactions to stockholders.
And while supporters of the Istanbul club may have to wait for official confirmation of the switch, Besiktas whetted their appetites with a photo of Pepe decked in the side's black and white shirt.
Pepe himself suggested the identity of his new club would soon be public knowledge with a teasing post on social media.
"Is this the season?", he wrote in Turkish, in a post accompanied by a picture of Besiktas' stadium.
The defender, who was born in Brazil but took up Portuguese nationality and debuted for his adopted country in 2007, enjoyed a sparkling decade at the Bernabeu after moving from Porto in that same year.
Madrid lifted 13 major titles during Pepe's time at the club, including an incredible three Champions League crowns in four years, although he was forced to be a spectator in June's rousing victory over Juventus to take the 2016-17 edition.
Pepe made a total of 334 appearances for the Merengue , netting on 15 occasions in all competitions.