The pair, vying for top spot in the league, go head-to-head this Sunday, then meet in the cup for the first time since Dirk Kuyt scored a late victor when Liverpool beat United 2-1 in January 2012.
They are now the top two in the Premier League.
Crawley Town's reward for their giant-killing of Leeds United is a trip to Bournemouth.
Holders Arsenal will visit either Southampton or Shrewsbury Town in round four, with that third-round tie yet to be decided after being called off when Shrewsbury reported Covid-19 cases in their squad.
The victors in the Manchester United v Liverpool showdown will face West Ham or Doncaster, while either Wycombe or Tottenham will travel to the winners of the Everton v Sheffield Wednesday tie.
United and Liverpool, who now occupy the top two spots in the Premier League, will face off for a place in the FA Cup fifth round at Old Trafford later this month (they also play each other at Anfield in the league this Sunday).
Reacting to the draw, Chorley manager Jamie Vermiglio admitted: "That's what we really wanted, Premier League opposition, great".
Not only does it allow Pep Guardiola's side an easy passage towards the final, but it also means he can rest players for the league, as well as cruising through matches without as much risk of injury and fatigue. City will take on fourth-tier Cheltenham and Tottenham plays Wycombe, which is in last place in the second-tier Championship.
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