Kompany, 33, played his final game for the club in Saturday’s 6-0 FA Cup final win over Watford.
The Belgian has played 360 games for City since joining from Hamburg in 2008, winning four Premier League titles, two FA Cups and four League Cups along the way – including one of each this season.
Manchester City say an announcement on Kompany’s future will be made imminently.
Commenting on his decision, Kompany wrote a letter to fans on Facebook, saying: “We’ve just seen the close of an incredible season. My 11th as a Blue. I cannot believe I’m writing this but … also my last as a Blue.
“Countless of times have I imagined this day, after all, the end has felt nearby for so many years. It still doesn’t feel real. Man City has given me everything.
“As overwhelming as it is, the time has come for me to go. And what a season to bow out. I feel nothing but gratefulness. I am grateful to all those who supported me on a special journey, at a very special club.
“I remember the first day, as clear as I see the last. I remember the boundless kindness I received from the people of Manchester.
“I will never forget how all Man City supporters remained loyal to me in good times and especially bad times. Against the odds you have always backed me and inspired me to never give up.
“Sheikh Mansour changed my life and that of all the City fans around the world, for that I am forever grateful. A blue nation has arisen and challenged the established order of things, I find that awesome.
“I cherish the counsel and leadership of a good human being, Khaldoon Al Mubarak. Man City could not be in better hands.”
Man City chairman Al Mubarak has paid tribute to Kompany’s huge contribution to the club’s success during his 11 years at the Etihad Stadium.
“There have been many important contributors to Manchester City’s renaissance, but arguably none are more important than Vincent Kompany.
“He defines the essence of the club. For a decade he has been the lifeblood, the soul, and beating heart of a supremely talented squad. A booming voice in the dressing room yet a quiet and measured ambassador out of it, Vincent can be as proud of himself as we are of him.
“I am not sure he expected to lift the Premier League trophy on four occasions during his captaincy but he will be remembered and revered whenever this period of unprecedented City success is spoken about by future generations.
“His leadership, intelligence and determination have seen him adapt brilliantly to playing under four different managers and overcome some debilitating injuries. He is a special character who has answered every demand the club has made of him.”