South Korean Son is reportedly on Newcastle's radar and his agent recently revealed the striker might consider his future at Imtech Arena at the end of the season.
However, Arnesen has stated he has no intention of letting the 19-year-old leave the club anytime soon.
"He is good. We won't sell him," he told Bild.
After a disappointing league campaign which saw Hamburg struggle in the bottom half of the table for most of the season, Arnesen is looking forward to next term and has lined up a number of transfer targets.
Bayer Leverkusen goalkeeper Rene Adler is set to sign on a free transfer and Latvian forward Artjoms Rudnevs is finalising his 3.5million euro (£2.8m) move from Lech Poznan.
Liverpool's Dirk Kuyt remains high on Arnesen's short-list but no contact has been made with the Premier League side.