According to the Turkish newspaper HaberTurk, Liverpool are keen to let Yeşil out on loan so that he can gain first team experience.
The 21-year-old was born in Germany but his family originate from the Trabzon region of Turkey.
Yeşil supported Trabzonspor growing up and was recently on holiday in his parents homeland.
The Reds striker recently attended Kadirga which is one of the largest festivals that takes place in Turkey. The event is held at the Tonya Yayla (plateau) in Trabzon – a large treeless mountaintop area.
Yeşil struggled for playing time this season due to a double cruciate ligament injury that ruled him out until the turn of the year. The young striker did however, make a full recovery and scored three goals for the U21 side since his return.