The Premier League's top four clubs meet in the world's oldest domestic cup competition this weekend while holders Wigan Athletic, now in the Championship, visit relegation-threatened Cardiff City on Saturday.
Chelsea and Arsenal are chasing an FA Cup-Premier League-Champions League treble while City can go one better as they will also contest the Capital One Cup final against Sunderland next month.
City host Chelsea on Saturday, their second meeting inside a fortnight after Mourinho produced a tactical masterclass to end the 100 percent home league record of Pellegrini's team this season.
Mourinho, disappointed to see his side concede a late goal to draw 1-1 at West Bromwich Albion in the league on Tuesday, has been in a war of words with Pellegrini over their respective title chances, Financial Fair Play, transfers and an off-the-ball incident involving Yaya Toure.
City, who had their game with Sunderland on Wednesday called off due to bad weather, have suffered a dip in form in recent weeks.
Despite being the fresher of the two sides, they will be without Brazil midfielder Fernandinho and Argentina striker Sergio Aguero against Chelsea.
Other than one outburst last week, City manager Pellegrini has tried to remain silent in the face of Mourinho's public pronouncements although he did acknowledge "the team with the strongest mentality is the one that will win the title".
Sunderland host Southampton on Saturday when Sheffield Wednesday also entertain fellow Championship side Charlton Athletic.
Arsenal, nine years without a trophy, could do with a boost to their fragile confidence when they host Liverpool on Sunday, having been blown away 5-1 when the teams met at Anfield last weekend.
A clean sheet in a 0-0 draw with Manchester United on Wednesday will at least have handed the Arsenal defenders a timely lift.
Defender Per Mertesacker said the Liverpool defeat was "one of the worst setbacks" of their season.
"Every defeat creates a difficult moment (but) it could have been worse," said manager Arsene Wenger.
"After (the opening-day defeat by) Aston Villa nobody would have put us in the position we are in today."
Liverpool captain Steven Gerrard's late winner from the penalty spot that secured a 3-2 victory at Fulham on Wednesday will add even more belief to Brendan Rodgers's side.
Also on Sunday, Everton host Swansea City and third-tier Sheffield United welcome Championship team Nottingham Forest.
Championship side Brighton and Hove Albion entertain Hull City on Monday.