Speaking to reporters after his side lost 2-1 to Arsenal at the Emirates Stadium on Monday, their fifth league defeat of the season, Pellegrini said that the return of key players was vital if they are to overtake the Gunners and surprise leaders Leicester City in 2016.
"We must improve because we are conceding too many goals," Pellegrini, whose side have only kept two clean sheets in their last 12 Premier League games, said.
"We hope that Vincent will return soon. If we have Vincent, David (Silva) and Sergio (Aguero) playing normally every week then I suppose that we are going to continue being an important team for the fight for the title."
While Aguero and Silva are important to City, Kompany appears to be a player they cannot do without.
They have conceded 18 goals in the nine matches they have played this season without Kompany, who has been out with a calf injury since September.
In the eight they have played with him, they conceded just one.
A defensive slip up from Eliaquim Mangala, who started in the centre of defence on Monday alongside Aleksandar Kolarov, handed Arsenal their second goal on the stroke of half time -- a deficit from which City could not recover.
"I think in the last games we made a lot of collective mistakes. Today we improved in that sense and we made an individual mistake," Pellegrini said.
City had more of the ball and created more scoring opportunities in Monday's match, which could have seen them leapfrog the London side into second place in the last match before Christmas, as Pellegrini pointed out.
"We dominated in possession and in options to score against a very good offensive team like Arsenal but of course if we concede two goals we must continue to try to think why," he said.
The loss ramps up the pressure on Pellegrini at a time when Pep Guardiola, who will leave Bayern Munich at the end of the season, has been linked with the club.
Asked if he found it difficult to rise above the swirling speculation, Pellegrini said: "It is not difficult, it is easy. You must be concentrated and focused on your work.
"I understand that we must improve, especially in the games we are playing away, but we continue to be involved in all the competitions."
(Editing by Martyn Herman)
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