The sporting director has admitted he could be forced to part company with the striker, with Dortmund at the head of the queue after anticipating an exit for Robert Lewandowski
The South Korean has been in fine form this season and has caught the eye of Borussia Dortmund, along with Premier League duo Liverpool and Tottenham.
Goal.com understands that Dortmund are set to beat Tottenham and Arsenal to the South Korean's signature as they prepare for a possible exit for Robert Lewandowski, and Arnesen is ready to let the youngster depart should he reject to commit his future to his current employers.
"If Son won't extend, we'll sell him," the former Chelsea sporting director told Sport Bild.
The 20-year-old joined Hamburg's youth academy at the age of 16 and his current deal with the Germans expires in June 2014.
Son has appeared in 71 Bundesliga matches for Hamburg, scoring 19 goals in the process.