The two sides, who are both top of the Premier League table and go head-to-head in the league on Monday (NZ time), meet in the cup for the first time since Dirk Kuyt scored a late winner as Liverpool beat United 2-1 in January 2012.
United have not beaten Liverpool since March 2018, with Ole Gunnar Solskjaer yet to taste victory against the Anfield side in four attempts as manager, once as Cardiff City boss.
Defending champions Arsenal will be on the road and play either Southampton or Shrewsbury Town, with the third round tie yet to be decided after the game was called off when Shrewsbury reported Covid-19 cases in their squad.
The FA said there will be a meeting of its Professional Game Board this week to decide on the fate of the Southampton v Shrewsbury Town fixture.
Although a rising number of Covid-19 infections - 36 in the last round of tests in the Premier League alone - has forced many to call for a break in the season, there is no let-up of fixtures as authorities look to complete all fixtures by May.
Manchester City travel to fourth tier side Cheltenham Town while Tottenham Hotspur, who beat eighth tier Marine, travel to second tier Wycombe Wanderers.
Chelsea host Championship side Luton Town while Sheffield Untied, who recorded their first win of the season at Bristol Rovers in the third round, host Plymouth Argyle.
Fulham take on Burnley in an all-Premier League matchup while League Two side Crawley Town, who upset Leeds United, are away at Bournemouth.
The draw for the fifth round was also made on Monday, with the Premier League's 'Big Six' avoiding match-ups against each other should they advance.
FOURTH ROUND DRAW (to be played weekend of Jan 23-24)
* Cheltenham Town v Manchester City
* Bournemouth v Crawley Town
* Swansea City v Nottingham Forest
* Manchester United v Liverpool
* Southampton/Shrewsbury Town v Arsenal
* Barnsley v Norwich City
* Chorley v Wolverhampton Wanderers
* Millwall v Bristol City
* Brighton & Hove Albion v Blackpool
* Wycombe Wanderers v Tottenham Hotspur
* Fulham v Burnley
* Sheffield United v Plymouth Argyle
* Chelsea v Luton Town
* Stockport County/West Ham United v Doncaster Rovers
* Brentford v Leicester City
* Everton v Sheffield Wednesday
FIFTH ROUND DRAW (to be played week starting Feb 8)
* Fulham or Burnley v Bournemouth or Crawley
* Manchester United or Liverpool v Stockport County/West Ham or Doncaster Rovers
* Sheffield United or Plymouth Argyle v Millwall or Bristol City
* Chorley or Wolves v Southampton/Shrewsbury Town or Arsenal
* Barnsley or Norwich City v Chelsea or Luton Town
* Everton or Sheffield Wednesday v Wycombe Wanderers or Tottenham Hotspur
* Swansea or Nottingham Forest v Cheltenham or Manchester City
* Brentford or Leicester v Brighton or Blackpool