Belgium central defender Kompany, 27, picked up the injury in the 4-0 win over Newcastle and it had been hoped he would return within two weeks.
"I prefer the doctor has the final word on this but I think he will be out for at least one month," confirmed boss Manuel Pellegrini.
"If we do strengthen (at centre back) it will not just be because of Vincent Kompany’s injury. We have a very good squad and we see what happens.
"Today or tomorrow we will know exactly how long Vincent will be out for."
City had already been without injured Serbia defender Matija Nastasic, although the club added he could return to fitness in time for the weekend match against Cardiff.
If Nastasic does not return, Joleon Lescott and Micah Richards would be the only specialist senior centre-halves at the club, although youngster Dedryck Boyata has first-team experience while Yaya Toure could be asked to play there.
The Premier League runners-up have been linked with Argentine defender Martin Demichelis, who only joined Atletico Madrid this summer.