The highly-rated youngster, who had previously been linked with Arsenal, boasts an impressive strike rate having scored 22 goals in 26 appearances for his country at under-16, under-17 and under-18 level.
Yesil took advice from former Liverpool captain Sami Hyypia, now the Leverkusen manager, before making the switch.
The club say he has been signed initially to bolster their youth ranks.
Yesil told the club's website, www.liverpoolfc.com: "I am really happy about this, it is my dream to play for Liverpool and I will give my all.
"I spoke with Sami Hyypia, he helped me. He told me very important things about Liverpool, the quality of Liverpool and the quality of life here, and how famous the club is."