League rules dictate caretaker coach Rodolfo Cardoso can only continue in the job until Tuesday, as the former Hamburg player does not have a coaching licence.
Despite initially indicated that he would be seeking an exception from the German Football League, Arnesen no longer appears keen on fighting Cardoso's corner.
Speaking after his side's 2-1 defeat to Schalke on Sunday, Arnesen confirmed the search is on for a new man to take over after the international break.
"Today or tomorrow, something has got to happen," Arnesen told Sky television.
"We don't have time until Christmas. For our next game in Freiburg, we will have a coach with a licence on the bench."
Former Hamburg player Sergei Barbarez has put himself into the frame for the job, saying: "I don't want to send a video application, but I just want to say that I have a very strong bond with this club and I have lots of emotions for this club.
"I think a solution with somebody from Hamburg would not be a bad one.
"I have had a small chat with Frank Arnesen and we will have to see what happens."
Hamburg are searching for a new coach after sacking Michael Oenning last month. Arnesen's preferred replacement, Huub Stevens, rejected Hamburg's overtures and instead joined Schalke last week.