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.