The 20-year-old's impressive form this season has reportedly attracted interest from a number of clubs in Europe, including Manchester United and Arsenal.
"Son, with his 20 years, is one of the players we would really like to keep hold of," Arnesen told NDR.
"We are willing to push ourselves to our pain barrier for him. He still has a year to go on his contract and we haven't had any offers from other clubs.
"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."
Son has been at Hamburg since 2008.