The 45-year-old is infuriated by what has been written in German newspaper Bild, hinting that he leaked information about the Portuguese tactician heading back to the Premier League.
"I heard this story before our game with Fortuna and I have to admit I was very sad and angry because it is completely false."
Klopp said that he regards Mourinho as a personal friend, and that he would never disclose sensitive or personal information, such as plans to return to Stamford Bridge.
"Mourinho is a colleague and a friend and I would never mess around in such a manner," he added. "It is a made up story."
The two managers come face-to-face again in their Champions League semi-final second leg clash at the Santiago Bernabeu on Tuesday, with Dortmund holding a 4-1 lead from last week's match, thanks to four goals from Robert Lewandowski.
The in-demand striker has put the Bundesliga club within touching distance of this year's Wembley final, but Mourinho has made clear Real Madrid still have a chance of turning round the deficit on home soil, and succeeding Chelsea as European champions.