The other tie sees Tottenham take on Brentford, with the Championship side in the first major semi-final of the club’s history.
That clash is also a derby, but the London clash may not have the same intensity of the Manchester derby between two of the top teams in the Premier League.
The semi-finals will be played in the week commencing 4 January 2021, with the ties played over single-legs instead of the usual two-legged affairs.
The final does not come round until 25 April at Wembley.
Tottenham saw off Championship outfit Stoke City on Wednesday, before Manchester United completed the line-up with two late goals giving them a 2-0 win over Everton at Goodison Park.
Carabao Cup semi-final draw
Manchester United v Manchester City
Tottenham v Brentford
Tottenham have not played Brentford since a clash in the same competition in 2000 when Spurs won 2-0.
The Bees have not beaten their London rivals since 1948 in a Division Two fixture.
City and United met in the semi-finals stage last season over two legs, with City winning the first leg 3-1 and progressing despite losing the second 0-1.