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 on Sunday afternoon, 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 on Saturday 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.
Wigan chairman Dave Whelan described a potential Wembley date with Blackburn as a "dream" - and revealed he may lead the team out himself.
As a player, Whelan played in the 1960 final for Rovers but saw his Wembley dream turn sour as a broken leg forced him off before half-time.
"The dream is there," he told Sky Sports News. "All Blackburn have got to do now is get through the replay and my dream is on, I can go to Wembley and (we can) play against Blackburn Rovers.
"When you've been to Wembley and got what I got, a broken leg and Rovers went on to lose 3-0, it's very upsetting. I feel I've got unfinished business at Wembley.
"Roberto's asked me to lead them out at Wembley and if we get Blackburn, I may just say yes."
McManaman is one of a host of fringe players to have excelled in the Latics' cup run and Whelan said: "Roberto's played his young boys - one of them, McManaman, scored a great goal on Saturday - so it's done us good all round.
"The performance they put on at Everton was stunning, in the first half we played Everton off the park and should've been four up."
The run to the last four contrasts with Wigan's seemingly annual struggle for Premier League survival, which the pragmatic Whelan admits has to remain their priority.
"We have to stay in the top flight, that's number one," he said. "But the FA Cup is still a fantastic trophy and we've never been to a final.
"To win the cup and stay in the league, that's the dream isn't it? And to get to the Europa League, that's unthinkable for Wigan Athletic. It would be wonderful."