The influential defender limped off during the second half of City's season-opening Barclays Premier League win over Newcastle on Monday.
The 27-year-old will miss City's next two matches against Cardiff and Hull and is also likely to be unavailable for Belgium's World Cup qualifiers in September.
Pellegrini said: "I prefer the doctor to say the last word, but I think (he will be out for) at least one month. The doctor can give more information."
Kompany's absence leaves Pellegrini hoping fellow central Matija Nastasic, who suffered an ankle injury last month, will be fit to return at Cardiff.
Nastasic was expected to miss the first month of the season but returned to training this week.
Pellegrini said: "We will see. Nastasic is working normally with the squad and we will see if he starts."
Kompany was sorely missed last season when a two-month lay-off with a calf injury badly damaged City's title hopes.
With Kolo Toure having joined Liverpool in the summer, other options in his absence, if Nastasic is not fit, are limited.
Right-back or centre-back Micah Richards is also injured and it was midfielder Javi Garcia who was deployed alongside Joleon Lescott in a makeshift back line in the closing stages on Monday.
Another midfielder, Gareth Barry, could also be deployed there while youngster Dedryck Boyata, who has had loan spells at Bolton and FC Twente in the past two seasons, might hope for a chance.
The situation will add to speculation that City could move into the transfer market before the September 2 deadline to strengthen their squad.
They have been linked with Real Madrid's Pepe and Atletico Madrid's Martin Demichelis this summer.
Pellegrini, who worked with Demichelis at Malaga, has not ruled out further additions but is giving little clue as to who he might like.
Speaking at a press conference to preview the Cardiff game, the Chilean said: "It is not dependent on what happens with Vincent Kompany.
"I told you last week that until the last day we will see what we can do but we have a very good squad so we will see what happens."
The injuries have given Lescott a chance to re-establish himself after falling down the pecking order under Roberto Mancini last season.
The England international is now in the final year of his contract at the Etihad Stadium and his long-term future is unclear.
Pellegrini did not want to disclose information about new discussions.
"We are not talking about those things," he said.