Both teams beat lower-league opposition to reach round four, with Unai Emery's Gunners winning 3-0 at Blackpool and United seeing off Reading 2-0 at home.
Arsenal - who have won the FA Cup a record 13 times - will play host to Ole Gunnar Solskjaer's rejuvenated Red Devils as the Premier League rivals meet in the competition for the first time since 2015.
Arsene Wenger's Arsenal came out on top in that sixth-round meeting at Old Trafford, but the man who lifted the FA Cup an unmatched seven times will not be in the dugout this time around, with Emery achieving a 2-2 draw away to Jose Mourinho's United before Christmas.
Elsewhere, holders Chelsea - who beat 12-time winners United in last season's final - will host the victors of a replay between Sheffield Wednesday and Luton.
There are two other all-Premier League ties as champions Manchester City welcome struggling Burnley to the Etihad Stadium and Crystal Palace host Tottenham in a London derby at Selhurst Park.
Barnet, the only non-league side left heading into the fourth round, were drawn at home to Sky Bet Championship side Brentford.
League Two pair Newport and Oldham failed to land plumb ties after seeing off Premier League sides Leicester and Fulham, respectively, in the previous round.
Newport's prize for beating the Foxes is a trip to the Riverside Stadium to take on Middlesbrough, while Oldham travel to Doncaster.
Wolves won the final match of the third round as they beat a much-changed Liverpool side 2-1 and they will now travel to either Shrewsbury or Stoke.
Marco Silva's Everton will take on 2004 finalists Millwall at the New Den, with AFC Wimbledon manager Wally Downes hosting West Ham, the Premier League side where he previously worked as a coach under Sam Allardyce.
There is the potential of another top-flight meeting, with Watford waiting for the winners of a replay between Premier League side Newcastle and Championship outfit Blackburn.
Brighton won comfortably at Bournemouth in round three and will welcome West Brom to the Amex Stadium, with the winners of a replay between Derby or Southampton travelling to face Accrington.
Meanwhile, Swansea play host to Gillingham, Bristol City take on Bolton and QPR travel to League One leaders Portsmouth.
FA Cup fourth-round draw
- Swansea v Gillingham
- AFC Wimbledon v West Ham
- Shrewsbury or Stoke v Wolves
- Millwall v Everton
- Brighton v West Brom
- Bristol City v Bolton
- Accrington v Derby or Southampton
- Doncaster v Oldham
- Chelsea v Sheffield Wednesday or Luton
- Newcastle or Blackburn v Watford
- Middlesbrough v Newport
- Manchester City v Burnley
- Barnet v Brentford
- Portsmouth v QPR
- Arsenal v Manchester United
- Crystal Palace v Tottenham
