The Red Devils will have to overcome Cambridge United in a replay after their dour goalless draw on Friday, and will play away in the fifth round if they progress.
The Gunners, who eventually saw off Brighton on Sunday, will now host Boro after Aitor Karanka's men picked up a shock 2-0 victory over Manchester City.
Liverpool, should they get past Bolton Wanderers in their replay, will face Crystal Palace away.
Bradford City, conquerors of Chelsea in remarkable circumstances, will face Sunderland or Fulham.
Elsewhere, there will be a Midlands derby as Aston Villa will host Leicester City, Blackburn Rovers meet Stoke or Rochdale, who play on Monday, West Brom face West Ham and Derby host Reading.
