The 27-year-old first moved to the Bundesliga side on loan in January 2013, with the deal extended for this season.
Djourou, who came through the Arsenal youth system, would be keen to stay on.
"I know the coach, I love the club, I love the city. HSV (Hamburg) is incredible," Djourou said in German newspaper Bild.
"The club has so much potential. We have to go way up.
"The fans are great. The mood is different than in England. If all goes well and the club also want to, I would stay."