The Manchester sides will avoid each other in the FA Cup quarterfinal round as United visit Wolverhampton, while City to travel to Championship side Swansea City.
Aside from United-Wolves, the other all-Premier League tie will be Watford hosting Crystal Palace.
Millwall will host Brighton & Hove Albion in the other match pitting a Championship team against a Premier League side.
The matches will take place between March 15-18 and can be streamed live in the U.S. on ESPN+.
United's tie with Wolves will be their third straight in the competition as visitors to a fellow Premier League side, after beating defending champions Chelsea 2-0 on Monday.
For City, they return to south Wales after defeating Newport last week.
The Manchester derby (in Premier League play), originally scheduled to take place March 17, will be postponed to a later date.
Overall, United have won the competition 12 times, one behind all-time leaders Arsenal. City have won it five times, and Wolves have four titles. The other contenders are seeking their first title.