He has backed interim boss Rodolfo Cardoso to hold the fort successfully in the meantime.
Arnesen's first target Huub Stevens, with whom he held talks over the weekend, joined Schalke, meaning HSV have to go back to the drawing board as they seek a successor to Michael Oenning, who was sacked last week.
"We have no defined timeframe," Arnesen told his club's website. "It is not the case that we desperately need to present a new coach before the Schalke game (on Sunday).
"I have always said that I wanted to talk to numerous candidates. After talking to Huub Stevens, I decided that it would not work out.
"Without wanting to go into any details about our talks, I can only say this much: I want a coach who is 100% committed to our club.
"That is the feeling that I need."
Arnesen is confident the team is in safe hands under Cardoso after witnessing the 2-1 victory at Stuttgart on Friday - the club's first win of the season.
"He (Cardoso) remains himself. He speaks logically, clearly and calmly. He doesn't say much, but what he does say is very good and to the point," added Arnesen.
"At half-time in Stuttgart, for example, he remained very calm and said that nothing was lost while we were 1-0 down.
"He then motivated the players and I was very impressed with the way he did that."
Cardoso has emerged as a candidate to land the job on a full-time basis although one potential obstacle could be the 43-year-old's lack of a valid coaching licence.
He is permitted to work as head coach for an interim period of only two weeks, according to German Football Association (DFB) and German Football League (DFL) rules, after which he must study to an obtain a licence.
However, Arnesen admits he has already made contact with the authorities to discuss whether an exception can be made.
"Rodolfo Cardoso currently has our 100% backing," he said. "The fact is that he cannot carry on working as coach following the Schalke game due to the lack of a coaching licence.
"We are looking for the best solution for HSV and maybe I can get a bit more time - we will look to talk with the DFB and the DFL."