According to Turkish tabloid Harberturk, Mikel is willing to accept Besiktas deal only if he is offered first team football.
Mikel Obi has been at Chelsea for 10 years but his time with the club could be up with Leicester midfielder N'Golo Kante and AS Roma star Radja Nainggolan heavily linked with the 2015 Premier League champions.
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According to Turkish tabloid Harberturk, the 29-year-old is willing to accept Besiktas deal only if he is offered first team football.
"Mikel did enquire about Besiktas and there were talks it all depends on what their board decide to do," one of his representatives, Mesrun Izzet is quoted to say by Haberturk.com.
"Money is of course important but Mikel wants first-team football if he is to move.“
Besiktas tried to get Mikel Obi last year but Jose Mourinho stopped the move as he wanted the midfielder to be part of his squad.
The Turkish side is also trying to lure another Nigerian player from Chelsea, Kenneth Omeruo.
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Omeruo joined Chelsea in 2012 but has spent all his time away from the club on loan.
Ahead of next season, Besiktas are willing to get the defender but are waiting for the permission of new Chelsea coach, Antonio Conte.