Sky Sports reports how the 20-year-old has been in impressive form in Germany this season, and is currently the club's leading goalscorer with five goals from 10 league appearances.
Gunners boss Arsene Wenger has been credited with an interest in bringing Son to the Emirates Stadium in January, as he looks to further strengthen his forward line in the forthcoming transfer window.
However, Arsenal could face competition from Premier League rivals Liverpool, with Brendan Rodgers also keen to add attacking reinforcements to his Anfield ranks, in a bid to improve the Reds' current standing.
Son is under contract at Hamburg until 2014, but after revealing last month that he is unsure whether he'll sign a new long-term contract to stay at the AOL Arena, speculation has sparked with regards to the Asian star's potential availability.
"We want to achieve something with Son," Fink told German Sky Sport News. "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."