The Turkish sports newspaper claim that the Black Eagles are looking for alternatives to Liverpool midfielder Lucas Leiva.
Beşiktaş were closely linked with a loan move for Leiva but the experienced midfielder is expected to stay at the club after making his first start of the season in the goalless draw against Arsenal at the Emirates on Monday 24 August.
Nigerian newspaper Punch recently reported that the Black Eagles offered Chelsea €5 million for Mikel.
Milliyet however, state that the Istanbul giants are interested in a loan transfer with a future buyout clause option.
Beşiktaş hope to bring in a replacement for Tolgay Arslan and Veli Kavlak as both midfielder are out injured until the turn of the year.
The Nigerian international has also been linked with Istanbul rivals Fenerbahçe and UAE side Al Ain over the summer.
The experienced midfielder is expected to struggle for playing time at Chelsea this season after falling down the pecking order at the London based club but he does have another two-years left on his contract.