The Super Lig outfit have lost seven of their last eight matches and are struggling off the park as well.
Turkish publication Fanatik claims that current manager Abdullah Avci could find himself out of work soon, and reports that Cebi's number one target is Gerrard.
The former Liverpool captain has built the Ibrox side into genuine title-contenders since arriving in the summer of 2018, and recently signed a new deal keeping him at the club until the summer of 2024.
Rangers are just two points shy of Old Firm rivals Celtic in the race for the Scottish Premiership title while both teams are in the knockout stages of the Europa League.
Fanatik claims that, despite Cebi's wishes, Besiktas would almost certainly be unable to fund a move for the former England internationalist and claims that he receives £1.6 million a year in salary.
The publication also believes that the coincidental timing of Reds boss Jurgen Klopp's contract expiring at the same time as Gerrard's Ibrox deal is a sign that the 40-year-old is viewed as the German's successor at Anfield.
It's not the first time Gerrard has been linked with another managerial post - Derby County were credited with an interest before Frank Lampard joined Chelsea.