Pep Guardiola's dethroned Premier League champions agreed a £40 million ($52 million) fee for the Dutch player, rising to a potential £41 million.
The signing comes a day after the club announced they had landed Spain Under-21s winger Ferran Torres from Valencia.
"City have been the best side in England over the course of the last decade," Ake, 25, said in a club statement.
"Coming here is a dream for me. This is a top side full of world-class players. Everywhere you look in this squad there are big names with international pedigree."
City director of football Txiki Begiristain said: "Nathan is a quality defender. He was a leader during his time at Bournemouth and I am so pleased we've managed to bring him here. He's strong and quick, and we feel he has the technical and intellectual attributes needed to excel in Pep's set-up."