Kompany limped out early on in the FA Cup fourth-round win at Stoke with a strain and missed the champions' goalless draw at QPR in the Premier League on Tuesday night.
Mancini will plan without the Belgium centre-back for the next few games and seems resigned to the fact his on-field leader will miss the FA Cup fifth-round tie with Leeds on 17 February.
That would make the home game against Chelsea on 24 February 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 Africa 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 fill in during Kompany's extended absence alongside Joleon Lescott.
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