There were no official details of the transfer fee but German media estimated it was 25 million euros ($28.40 million).
The gifted 23-year-old Calhanoglu has not played since February after the Court of Arbitration for Sport (CAS) upheld a four-month ban imposed by world governing body FIFA over his contract dispute with Turkish club Trabzonspor from 2011.
Trabzonspor lodged a claim alleging Calhanoglu had breached the terms of his contract without just cause when he signed for Karlsruhe in 2011. Calhanoglu was a minor at the time.
Calhanoglu was ordered to pay Trabzonspor 100,0000 euros ($1.14 million). He joined Hamburg from Karlsruhe in 2012 and moved to Leverkusen two years later.
"The three years at Leverkusen were very emotional, Calhanoglu said in a club statement. "The support was outstanding especially during my four-month ban."
"Now I want to start a new chapter in my career."
($1 = 0.8803 euros)
($1 = 0.8804 euros)
(Reporting by Karolos Grohmann; Editing by Ken Ferris)
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