The draw, conducted at Wembley immediately after United's 2-2 draw with Chelsea at Old Trafford, pitched the eventual winners of the replay against City. Millwall and Blackburn also drew earlier in the day and will replay on Wednesday, with the winner to face Wigan.
The semi-finals will be played at the national stadium over the weekend of April 13 and 14.
Chelsea boss Rafael Benitez was given the first chance to respond to the eye-catching draw, but was unwilling to look any further ahead than his side's replay against United.
"We have to think about the first game and, afterwards, think about City," he told ITV1. "It will be tough, will be difficult, United are a great side."
Javier Hernandez and Wayne Rooney gave United a 2-0 lead inside 11 minutes at Old Trafford, but well-taken second-half goals from Eden Hazard and Ramires earned Chelsea a replay.
City, in contrast, brushed off npower Championship side Barnsley 5-0 on Saturday as Carlos Tevez shrugged off his latest off-field controversy to score a hat-trick and set up his side's other two goals.
Wigan produced arguably the performance of quarter-final weekend to beat Everton 3-0 with goals from Maynor Figueroa, Callum McManaman and Jordi Gomez in less than four minutes in the first half.
Millwall and Blackburn played out a goalless draw at the Den, Andy Keogh hitting the post and Danny Shittu stretching keeper Jake Kean with a 40-yard bolt from the blue as the hosts created the better chances.
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