The daily Liverpool based paper state that the 22-year-old is out of contract and has not been offered a new deal by Reds.
Yesil joined Liverpool back in 2012 from Bayer Leverkusen, and was tipped to be one of the brightest young prospects around at the time.
The young striker went onto make just two first-team appearances for Liverpool after two serious knee injuries.
Yesil has been closely linked with a move to the Trabzonspor for the past four seasons.
“I have received offers from Trabzonspor for the past four years. I would like to join the club for the 50th year anniversary,” Yesil was quoted as saying by Super Haber.
The former Bayer Leverkusen player is known to be a Trabzonspor fan and spent his summer break in the Black sea region of Turkey.
Trabzonspor are looking to re-build their squad for the upcoming season following a disappointing year. The 2016-17 season will be Trabzonspor’s 50th year anniversary and the clubs President is targeting the Turkish Super Lig title.
The 22-year-old has made two assists and has found the back of the four times in 17 appearances for Luzern this season.