Sky Sports reports how negotiations between the Bundesliga club and the 20-year-old forward have reached an impasse, leaving his long-term future shrouded in doubt.
Gunners boss Arsene Wenger has been credited with an interest in bringing the South Korean international to the Emirates Stadium this summer, but will face stiff competition for the player's signature from north London rivals Spurs.
Andre Villas-Boas is also understood to be monitoring Son's situation as he weighs up a bid to lure the Hamburg star to White Hart Lane ahead of the 2013/14 campaign.
Son is under contract in Germany until 2014, but with just over 12 months remaining on his existing deal the club are conscious of losing one of their prized assets for nothing next year, and so could look to sell him when the transfer window reopens this summer.
With growing debts crippling Hamburg it will be important for the club to cash-in should it become clear that Son has no intention of committing his future to the Imtech Arena.
"If Heung-min Son is not signing a new contract and makes up his mind to definitely leave, and given a concrete offer from an interested club, then we would be forced to sell," Fink confirmed in an interview with Sport-BILD weekly.
"In a situation like this we would be unable to do anything else as we would be unable to keep hold of him."