Vettel is closing in on a hat-trick of drivers' titles and can extend his lead in the championship over Ferrari's Fernando Alonso at this weekend's Abu Dhabi Grand Prix.
The 25-year-old has won the last four races and led from start to finish in the last three, and has also been linked with a switch to Ferrari, a move the German has said is "bulls**t."
"Sebastian's grown up within Red Bull since he was 12. He's had all his success with Red Bull and is extremely comfy within the team," The Sun quoted Horner, as saying.
Horner is confident of keeping hold of his star driver, who he says will only get better.
"His agreements are fixed with us and I can't see any reason why he'd want to be anywhere else," he said.
"He's only 25 and I can see him being with us another five years. He'll only get better too. That's the exciting thing about him. He's driving better this year compared to last year because his experience is growing," he added.