The center back picked up the injury against Newcastle on Monday, while Manuel Pellegrini has reiterated that he could sign another defender before the transfer window ends.
The center back pulled up in the second half of the 4-0 victory over Newcastle United on Monday and had to be substituted. City boss Manuel Pellegrini has hinted the club could still strengthen the squad before the transfer window closes but will not let the injury to Kompany dictate his transfer activity.
"I prefer the doctor has the final word on this, but I think he will be out for at least one month," Pellegrini told reporters. "If we do strengthen [at center 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 is said to be interested in Atletico Madrid's Martin Demichelis as a potential center back target.