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.
The Belgium international will miss his country's World Cup qualifier against Scotland on September 6, as well as Premier League matches against Cardiff City, Hull City and Stoke City.
He is also a doubt for the first Manchester derby of the season on September 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."
(Reporting by Toby Davis; Editing by John Mehaffey)
