Kompany limped off after 71 minutes of Monday night's 4-0 win against Newcastle.
And boss Manuel Pellegrini says the side now face a waiting game before the 27-year-old returns to action.
"I prefer the doctor has the final word on this but I think he will be out for at least one month," said 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."
Kompany looks set to miss City's next two fixtures against Cardiff and Hull and is likely to be unavailable for Belgium's World Cup qualifiers next month.
The skipper's absence leaves City hoping Matija Nastasic will be fit to face Cardiff after suffering an ankle injury last month.
Despite expecting to miss the first month of the season, Nastasic returned to training this week.
Pellegrini said: "We will see. Nastasic is working normally with the squad and we will see if he starts."