The Germany youth international underwent a medical on Thursday and will join up with the Reds' youth team.
Yesil had not made a breakthrough into Sami Hyypia's Leverkusen first team but has scored 19 goals in 21 appearances for Germany's Under-17 side.
The striker told Liverpool's official website: "The first thing is that I have to train and then in one or two years I hope, step by step, to be in the first team.
"It is my dream to play for Liverpool and I will give my all."
Yesil won the silver shoe at last summer's Under-17 World Cup, scoring six goals and helping Germany to the semi-finals of the tournament, where they eventually lost to hosts and eventual winners Mexico.