The draw features three of the Cup’s four most successful sides, with Arsenal the record 13-time winners.
United, who beat Norwich to reach the last four, have won it 12 times, while Chelsea, who edged past Leicester in the quarter-finals, have eight wins.
Both ties will be played at Wembley on the weekend of 18 and 19 July. The final is scheduled for Saturday, 1 August.
City booked their clash with the Gunners after a Kevin De Bruyne penalty and Raheem Sterling goal saw off Newcastle on Sunday evening.
City faced Arsenal in a Wembley semi-final three years ago, where Arsene Wenger’s side managed to prevail over Pep Guardiola before going on to lift the cup.
But City have been rampant against Arsenal in recent years, and beat Mikel Arteta’s side 3-0 when they faced each other 10 days ago in the first game back from the coronavirus hiatus.
The clash between United and Chelsea is a repeat of the FA Cup final from two years ago, where Eden Hazard’s penalty gave the Blues victory.