The Nigerian international has been linked with a move away from Chelsea this summer - and speculation has increased since Jose Mourinho returned for his second spell as Stamford Bridge boss.
Chelsea have since signed Marco van Ginkel from Vitesse Arnhem, and the 26-year-old Mikel is now believed to be surplus to requirements.
Napoli, managed by former-Chelsea boss Rafael Benitez, are also keen on the defensive midfielder, and may consider a bid of their own following Galatasaray's move.
The Turkish giants are keen to strengthen and build on their Champions League success from last season. The double capture of Drogba and Wesley Sneijder helped Galatasaray to the Champions League quarter-finals last term.
Mikel has not featured in either of Chelsea's Premier League games so far this season, with Mourinho favouring Frank Lampard and Ramires.
And Chelsea have got off to a strong start under their new manager, winning their opening two games of the season against newly promoted Hull City on Sunday and Aston Villa on Wednesday evening.