The great rivals are the highest-placed sides left in the competition after Southampton shocked Arsenal on Wednesday to earn a two-legged tie with Jurgen Klopp's side.
United will play Hull City home and away for a place at Wembley.
Jose Mourinho's men required a stoppage-time winner from Marcus Rashford earlier this season to see off the Tigers, who reached a first League Cup semi-final in club history by beating Newcastle United on penalties on Tuesday.
Liverpool and Southampton met in this competition last season, with the Anfield side thrashing the Saints 6-1 at St Mary's Stadium, where the first leg in that tie will be held.
The first legs of the ties will be held on the week commencing January 9, with the following games a fortnight later.