United, who overcame City 3-2 at the Etihad Stadium, will face the Reds on the weekend of January 28 and 29.
United have won the competition 11 times, while Kenny Dalglish's men have triumphed on seven occasions.
The game will be the first meeting of the two sides since the controversial clash in October when Liverpool forward Luis Suarez racially abused United defender Patrice Evra. Liverpool accepted an eight-match ban for the Uruguayan over the incident earlier this month.
Here are some memorable past meetings between the sides in the competition.
Manchester United 1-0 Liverpool (third round, January 9, 2011)
Kenny Dalglish's second coming as Reds boss got off to the worst possible start as Ryan Giggs' second-minute penalty proved decisive for United, while Liverpool also had Steven Gerrard sent off.
Liverpool 1-0 Manchester United (fifth round, February 18, 2006)
A Peter Crouch goal in the 19th minute helped Liverpool to victory en route to winning the competition, beating West Ham in the final at the Millennium Stadium.
Manchester United 2-1 Liverpool (fourth round, January 24, 1999)
Liverpool almost ended United's treble quest prematurely when Michael Owen put them ahead in the third minute. But the Red Devils hit back with goals in the 88th and 90th minutes from Dwight Yorke and Ole Gunnar Solskjaer. The Norwegian sealed an even more famous late comeback in the Champions League final later that year to complete the treble.
Manchester United 1-0 Liverpool (final, May 11, 1996)
A forgettable match was decided by Eric Cantona's late strike after a poor punch from Liverpool goalkeeper David James. Indeed, it is the garish white suits worn by Liverpool in the pre-match walkabout for which this game is best remembered.
Manchester United 2-1 Liverpool (final, May 21, 1977)
All the goal action was concentrated into a five-minute spell early in the second half. Stuart Pearson put United 1-0 up in the 50th minute before Jimmy Case hit back for Liverpool two minutes later. However, Jimmy Greenhoff restored United's advantage, and they saw it out to claim the cup for a fourth time.
FA Cup head-to-head record: Played 16: Liverpool wins: 3 Draws: 4 Manchester United wins: 9.