The 20-year-old moved to the Imtech Arena from Karlsruher last year, and scored 11 goals in 32 Bundesliga appearances as Hamburg only avoided relegation by virtue of a play-off victory over Greuther Furth.
Calhangolu's 11 strikes included an incredible 50-yard swerving free-kick in a 3-0 win over Borussia Dortmund in February, and his impressive performances in a struggling side have led to speculation of a move away from Hamburg.
And Leverkusen chief executive Michael Schade claimed on Monday that an agreement has been reached between his club and the player.
"Calhanoglu should become our marquee signing," he told Bild.
"We have presented an offer in writing to HSV.
"We have an understanding with the player. He has verbally accepted our offer and only wants to join us.
"He has informed Hamburg of this."
