The Turkish Super Lig side are understood to be in talks with the 29-year-old forward, who has been linked with a reunion at Ibrox with former Anfield colleague Steven Gerrard while Hoops boss Neil Lennon is also believed to be keen on the player as he looks to beef up his attacking options.
Reports in Turkey, specifically in the Fanatik newspaper, suggest that Trabzonspor are closing in on a deal for the attacker, who has also played for Manchester City, Chelsea, Bolton and West Brom.
Ünal Karaman's side are believed to be offering Sturridge a deal worth £1.83 million a year which, it's claimed, would remove both Glasgow teams from the equation.
Fanatik highlights Gerrard's interest but an Ibrox switch looks increasingly unlikely if reports in Turkey are accurate.
Sturridge netted four times in 27 appearances for Jurgen Klopp's side last term.