Pep Guardiola's top-flight champions eased past Bristol City 3-0 on Wednesday (AEDT), while United backed up its Carabao Cup success with a 3-1 comeback victory over West Ham.
City will face former defender Kompany, who lifted four Premier League titles at Etihad Stadium, after being drawn at home to runaway Championship leader Burnley following its late 1-0 win against Fleetwood Town on Thursday (AEDT).
United ended a six-year trophy drought on Monday by defeating Newcastle United and will continue its quest for further silverware when it hosts Fulham at Old Trafford.
Fourth-tier Grimsby Town remain the shock package in this season's FA Cup after Premier League Southampton fell to a 2-1 home loss and became the Mariners' fifth victim from a higher division in the 2022-2023 campaign.
The League Two side's reward will be a trip to in-form Brighton and Hove Albion, which edged past Stoke City 1-0 in the fifth round on Tuesday.
Sheffield United was another surprise winner in the last 16 of the competition, defeating Tottenham 1-0, and will host Blackburn Rovers, which ensures at least one Championship side will reach the semi-finals.
FA Cup quarter-final draw in full:
Manchester City v Burnley
Manchester United v Fulham
Brighton and Hove Albion v Grimsby Town
Sheffield United v Blackburn Rovers
