Arsenal with 13 FA Cup wins and United with 12 are the most successful clubs in the world’s oldest knockout competition.
Premier League champions Manchester City are at home to top-flight strugglers Burnley, while holders Chelsea will take on Sheffield Wednesday or Luton Town at Stamford Bridge.
Giant killers Newport County of League Two (fourth tier), who beat Leicester City in the third round, face a trip to second-tier Middlesbrough, managed by Newport-born Tony Pulis.
Wolverhampton Wanderers, who knocked out Premier League leaders Liverpool on Monday, will be away to the winners of the replay between Shrewsbury Town and Stoke City.
Everton face a tricky test at Championship (second tier) side Millwall, while Crystal Palace will host Tottenham Hotspur in an all-London tie.
Non-league Barnet – the lowest-ranked club left in the competition — welcome Championship side Brentford.
Fourth round draw
Swansea City v Gillingham
AFC Wimbledon v West Ham United
Shrewsbury Town or Stoke City v Wolverhampton Wanderers
Millwall v Everton
Brighton Hove Albion v West Bromwich Albion
Bristol City v Bolton Wanderers
Accrington Stanley v Derby County or Southampton
Doncaster Rovers v Oldham Athletic
Chelsea v Sheffield Wednesday or Luton Town
Newcastle United or Blackburn Rovers v Watford
Middlesbrough v Newport County
Manchester City v Burnley
Barnet v Brentford
Portsmouth v Queens Park Rangers
Arsenal v Manchester United
Crystal Palace v Tottenham Hotspur
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