The Belgium international suffered the problem in Saturday's 3-1 win over Everton and is set to miss a number of crucial games in the coming weeks.
Kompany was due to link up with the Belgium squad ahead of Friday's World Cup qualifying game in Croatia, but he has been forced to withdraw.
The 27-year-old, a key player in City's backline, could also miss Premier League games against West Ham, Chelsea and Newcastle, and the Champions League clash in CSKA Moscow.
City suffered a shock defeat in Cardiff while Kompany was sidelined with a groin problem earlier this season and his absence will be a concern to manager Manuel Pellegrini after some erratic defensive displays by his team.
Serbian defender Matija Nastasic replaced Kompany at the weekend and is likely to deputise during his injury lay-off.
The goals in the Cardiff game were blamed on his absence (by the press), but the Villa game showed that our current defensive issues go beyond one players presence. I think we'd actually conceded around the same number of Premiership goals at the same stage last year and still ended up with the tightest defense, so I'm confident a few tweaks will sort things out and either Richards or Lescott (although he is very left sided) would be suitable replacements. So, as long as we keep Garcia out of the back four we'll be fine.- bernsteinforpm