The 47-year-old revealed this week that the Hoops had sponsored him to complete a coaching course and said being linked with succeeding Neil Lennon at Parkhead was an honour.
However, German publication BILD reports that the former New York Red Bulls manager, who is under contract with the Austrians until summer 2022, has made contact with the Celtic board to rule himself out.
Christian Falk, BILD’s head of football, wrote on social media saying: “I have received the confirmation: Jesse Marsch won’t join Celtic.”
Falk had earlier tweeted that he understood Marsch had told the Hoops he would not be available in the summer.
Speaking to BBC Scotland Marsch said: “Right now my full concentration is on finishing my year here in Salzburg.
"Then, I'll evaluate in the summer what the possibilities might be for a potential next step.”