Kompany helped City to 10 trophies, including this year's domestic treble, during 11 years and 360 games at the club.
The 33-year-old has been appointed as play-manager of Belgian side Anderlecht, a club he joined at the age of six.
Fellow Belgian De Bruyne was among those to take to Twitter to express their feelings.
He wrote: "Playing for about 10 years with this man for club and country. And what a privilege it's been.
"Big player, big personality and big leader. Learned a lot from you. Wishing you all the best for the future."
Kompany signed off his City career with a second FA Cup trophy on Saturday, where City thrashed Watford 6-0 to complete the treble.
Young midfielder Foden said his former team-mate had "looked after him from day one".
Kompany scored 20 goals during his 360 games, and won four Premier League titles and even those from other clubs praised the centre-back, including former Manchester United defender Gary Neville, ex-Liverpool striker Michael Owen and Match of the Day presenter Gary Lineker.
Neville admitted he wished Kompany had played for his club and said his influence on and off the pitch was "huge".
Lifelong City fan Liam Gallagher wrote: "Thankyou Vincent Kompany for everything good luck see ya soon LG x".
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