Kompany has been a hot cake since masterminding Burnley's swift return to the top flight at the first time of asking.
The former Manchester City captain has been linked with the Tottenham Hotspurs and Chelsea managerial jobs while a return to the Etihad Stadium as a potential successor to Pep Guardiola has been mooted.
Kompany is however staying put at Turf Moor after signing a new five-year contract.
Burnley secured promotion to the Premier League with seven games to spare and can reach 100 points with victory over Cardiff City in their final championship game of the season on Monday.
Kompany said: "Burnley and Turf Moor felt right pretty much from the start - so it also feels right to sign for the next five years.
"Together with the fans we have made Turf Moor a fortress again and continue to look forward to the future and the job of making Burnley better with every step."
Burnley chairman Alan Pace added: "We are ecstatic Vincent has signed a new deal. He is the perfect fit for how we all see Burnley Football Club moving forward.
"An extraordinary leader, setting ever higher standards and driving our club to the levels we all want to achieve."
As a player, Kompany joined Man City from Hamburg in 2008 and starred for eleven seasons, making 323 appearances and captaining the side to four Premier League titles, two FA Cups and four League Cups.
