The report states that Schalke general manager Horst Heldt confirmed that the Bundesliga side had ended their pursuit of the Switzerland international.
The Milliyet also claim that Besiktas have tabled an increased offer for the 31-year-old.
Leicester City, West Ham, Fiorentina, Hamburg and Werder Bremen are all also in the race to sign Inler according to Sky Sports journalist Gianluca di Marzio.
The Foxes newly appointed manager Claudio Ranieri revealed last week that he was close to securing a replacement for Esteban Cambiasso.
Cambiasso opted against extending his contract with the Foxes and departed the club at the end of last season.
Inler scored one goal in 33 appearances for Napoli in all competitions last season.
The Switzerland international has also previously played for Udinese, Zurich and FC Basel.