The pairing was announced after Wednesday night’s fifth round (round of 16) fixtures.
To get to the last eight, United had to rely on an 89th minute goal by Casemiro to beat Nottingham Forest 1-0 away.
While Liverpool overcame Championship side Southampton 3-0 at Anfield to progress.
FA Cup holders Manchester City will welcome Newcastle United at the Etihad and Wolverhampton Wanderers will entertain Coventry City.
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The quarter-final ties will be scheduled across the weekend of Saturday 16 March 2024.
FA Cup quarter-final draw:
Wolves vs Coventry City
Man United vs Liverpool
Chelsea vs Leicester City
Man City vs Newcastle United
