The 83-year-old billionaire, who has been heavily involved in the sport for more than 50 years, is currently fighting two court cases related to the sale of Formula 1 to private investors and has been heavily linked with an exit from his current role.
However, speaking at a Q&A session at the team's Milton Keynes headquarters, Horner suggested that he had no interest in Ecclestone's role and wished to continue with Red Bull.
He said: "I have no interest in the role he performs. My focus is in running a team.
"To replace Bernie would be impossible – certainly with one person so I am very happy with what I do here."