Freund had been heavily linked with a move to Stamford Bridge as part of Chelsea's co-controlling owner Todd Boehly's revamp of the club's administration.
It had been thought the London outfit were hopeful of making an appointment imminently but that would now appear unlikely.
Quoted in a statement issued by the Austrian club, Freund said: "As I have already confirmed in interviews, Chelsea FC were interested in me.
"When such a big club asks, it not only honours me and the work of FC Red Bull Salzburg, it is of course also a circumstance that entails personal considerations.
"But I've come to the conclusion that I'm in the best of hands with FC Red Bull Salzburg and that a change is out of the question for me."
Freund had been considered a good fit for Chelsea by Blues chiefs, who were impressed by his shrewdness, bold ideas and passion.
They felt that, after catching the eye at Salzburg, Freund would have dovetailed well with the new coaching staff headed up by recently-appointed manager Graham Potter.
It is believed Chelsea have not completely given up but they are also looking at other options. Mitchell, of Monaco, is also thought to be of interest.
Boehly is currently doing the job himself on a temporary basis.
Underlining Salzburg's confidence that Freund will be staying, their managing director Stephan Reiter said: "Christoph Freund has been doing excellent work for us for many years and only recently extended his contract until 2026.
"I can hereby confirm that he will remain sporting director of FC Red Bull Salzburg.
"Of course, he informed me about Chelsea FC's interest in him and we exchanged ideas on a personal level and in good conversations.
"The fact is that he will remain our sporting director. We have not received any inquiries about a possible change."