Calhanoglu has enjoyed an impressive breakthrough season with Hamburg, scoring 11 goals in 32 Bundesliga games.
The Turkey international signed a new contract to keep him at the Imtech Arena until 2018 earlier this year.
However, with Hamburg facing a relegation play-off with Greuther Furth in order to preserve their place in the German top flight, Calhanoglu's future has been cast into doubt.
And Leverkusen are front-runners to secure the 20-year-old's services according to representative Bektas Demirtas - who ruled out his client moving to German champions Bayern Munich or any interested Premier League clubs.
"From the perspective of the player, the environment at Leverkusen is favoured," Demirtas told Bild. "It would be the next step for Hakan to play internationally.
"They are a club who are playing in Europe, who would like to buy him and who are also prepared to pay the right price for him.
"A move to a foreign club is not in question. Bayern also plays no role, we have never favoured a move to Munich.
"It is important that the boy is playing. We have to think in the medium to long term."
The first leg of Hamburg's play-off with Furth takes place on Thursday.
