Kompany, whose imposing presence will be sorely missed at the heart of the City defence, limped off the pitch 71 minutes into their 4-0 victory.
He is also a doubt for the first Manchester derby of the season on Sept. 22.
"I prefer the doctor has the final word on this but I think he will be out for at least one month," Pellegrini said on the club website (www.mcfc.co.uk).
The City boss did not rule out making a late move in the transfer window to bring in defensive reinforcements.
"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," he added.
"Today or tomorrow we will know exactly how long Vincent will be out for."
- Sports & Recreation
- Cardiff City
- Hull City
- Stoke City