The club captain has been out since his side's win against Stoke City in the previous round, while the England midfielder is also a fitness doubt
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The Belgian subsequently missed his side's next three Premier League games and Mancini confirmed at a press conference that he is unlikely to play a part in Sunday's fixture.
Barry, meanwhile, played in the Premier League champions' last game – a 3-1 defeat at Southampton – but is also a fitness doubt with a knock.
However, Mancini stated that, injuries aside, he will play the best team available to him against Neil Warnock's men.
City won the FA Cup in 2011 and Argentine defender Pablo Zabaleta has urged his side to use the tournament to bounce back from their recent travails.