Kompany missed last weekend's 3-1 win at Newcastle United after limping off in the 21st minute of the derby following which the 26-year-old saw a specialist.
If the injury does prove more serious than City first thought then the concern for Mancini will be whether he recovers before Kolo Touré departs to play for Ivory Coast in the Cup of African Nations after the FA Cup third round tie against Watford on 5 January.
The absence of Touré, currently Mancini's favoured first reserve centre-back, plus Kompany would leave the manager with only two recognised defenders in the position, Matija Nastasic, who is in his debut season, and Joleon Lescott, now the fourth choice.