The former Dongbuk High School FC starlet has casted doubt over his future by not signing an extension to his deal at the Imtech Arena, which is set to expire in 2014.
Son has already scored five goals in 10 league appearances for Hamburg this season, and Fink is anxious to ensure that he does not lose him in January.
"We want to achieve something with Son,” Fink told Germany’s Sky Sports.
"I think it would be better for his sporting development, if he stayed and played with us.
"When anyone plays well, there is of course always a bit of transfer speculation about him.
"Ultimately, Son must know that we want to build something here and we have kept hold of players who want to also achieve something."