The Switzerland international, who previously spent time on loan at Bundesliga rivals Hannover, has made 14 top-flight appearances since arriving at the Imtech Arena in the close-season.
And, despite Hamburg's lowly position in the league, Djourou has a new feeling of optimism under new coach Mirko Slomka, under whom he worked at Hannover.
After seven consecutive league defeats, Hamburg began Slomka's reign in superb fashion by thrashing Borussia Dortmund 3-0 on Saturday and Djourou believes they have potential to do even better.
"I know the coach, I love the club, I love the city," he is quoted by Bild. "HSV is incredible. 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."
Djourou is likely to be a member of the Switzerland squad at the 2014 FIFA World Cup in Brazil, where the country line up in Group E alongside France, Ecuador and Honduras.
