The Belgium defender has missed five games since suffering a calf injury in the FA Cup fourth-round win at Stoke just over a month ago.
It had been hoped the 26-year-old could return against Chelsea last weekend but the problem has lasted longer than expected and Kompany has also had trouble in his other calf.
Mancini has not yet ruled him out for the trip to Villa Park but the cup quarter-final against Barnsley is now looking a more realistic comeback target.
Mancini said: "Vinny is getting better in this moment.
"We hope we can recover him for the next game. We will try for Aston Villa for Monday but it is difficult because he did not train with the team, he trained with the physio.
"We hope for the FA Cup.
"The calf is a difficult injury because you think you are okay and then you have a problem. The calf is a strange injury."
Former Villa midfielder Gareth Barry should be fit to face his old team after two games out with an ankle injury.
Mancini said: "He has trained with the team the last two days and I think he will be ready."