Son bagged a brace in Hamburg's 2-1 Bundesliga win over Mainz on Saturday, taking his tally to 11 goals in 28 appearances this term and 19 in 68 games for the German club since he arrived from FC Seoul in 2008.
His current contract has just over a year left to run and Arnesen says he is keen to hand the 20-year-old a new deal as interest grows in the South Korea international.
"Son, with his 20 years, is one of the players we would really like to keep hold of," he told NDR television.
"We are willing to push ourselves to our pain barrier for him."
Arnesen confirmed that contract negotiations are already at an advanced stage for the forward, who is attracting interest from clubs around Europe, even though no official approaches have been made.
"He still has a year to go on his contract and we haven't had any offers from other clubs," said Arnsesen
"We are at a good stage with him and his contract extension so I am positive that we will extend with him for the long term.
"We have to ensure that we extend his contract as quickly as possible so that he can be settled."