The 40-year-old has already won domestic honours with Bayern and was named Germany's Coach of the Year on Sunday. His next major target is the Champions League.
Bayern were knocked out by Paris Saint-Germain in last season's semi-finals, but Kompany believes his side can go further this time.
"We have big goals," the Belgian said as Bayern prepare to begin their Bundesliga campaign against Stuttgart on Friday.
Kompany arrived in Munich in 2024 after a difficult Premier League season with Burnley. His reputation had taken a hit after Burnley were relegated, despite having previously guided them to the Championship title with 101 points.
Bayern nevertheless decided to take a chance on their former defender after failing to land several other candidates.
The decision has proved successful. Kompany has not only delivered trophies but has also built a strong relationship with his players while avoiding the political problems that affected several of his predecessors.
Bayern sporting director Christoph Freund, who helped bring Kompany to the club, has been impressed by the manager's leadership.
"He's a good person and an outstanding personality," Freund said. "He is a leader and you can feel this every day. And he's a really hard worker.
"It's really fun to work with him every day. The team like him a lot."
Former Bayern coach and current Borussia Dortmund boss Niko Kovac also believes Kompany has shown he can handle the pressure of managing a dressing room packed with top players.
"He won a lot of titles, so he knows exactly how a dressing room is," Kovac said.
"I think Vincent understands exactly how to manage superstars."
Kovac also believes Kompany's relegation with Burnley should no longer define his coaching career.
"OK, he was relegated, but now he won titles, he reached the semi-finals in the Champions League, so what does relegation actually mean?" he said.
Bayern will begin their Bundesliga title defence at home to Stuttgart at the Allianz Arena on Friday, with Kompany already looking beyond domestic success as he aims to take the German giants back to the top of Europe.
