The 27-year-old limped off after 34 minutes against Everton on Saturday and has subsequently been ruled out of Belgium's World Cup qualifiers against Croatia and Wales
The 27-year-old limped off after 34 minutes at the Etihad Stadium and tests on Monday have subsequently revealed the extent of the defender's injury.
City have not said how long Kompany will be sidelined but he has been ruled out of Belgium's World Cup qualifiers against Croatia on Friday and Wales next Tuesday.
Elsewhere for City, Martin Demichelis has stepped up his recovery from a knee problem and has resumed light training during the international break.
The 32-year-old was signed by City from Atletico Madrid at the end of the summer transfer window but has yet to feature for Manuel Pellegrini's side.