Kompany limped out of last week's FA Cup fourth-round win at Stoke with a calf strain and missed the champions' goalless draw at QPR in the Premier League on Tuesday.
The Belgium centre-back will definitely miss Sunday's clash with Liverpool and next week's trip to Southampton and Mancini already seems to have ruled him out of the FA Cup fifth-round tie with Leeds on February 17.
That would make the home game against Chelsea on February 24 the most likely comeback date.
Mancini said: "Vinny is out. I hope he can recover after the FA Cup."
City were short on central defensive options at QPR with Kolo Toure at the African Cup of Nations and Matija Nastasic only able to make the bench due to a knee problem.
Midfielder Javi Garcia filled in at the back but Mancini is hopeful Nastasic will be fit to start against Liverpool at the Etihad Stadium alongside Joleon Lescott.