The draw for the last-eight took place at Deepdale just before Louis van Gaal's men beat Preston 3-1 in the final game of fifth-round weekend.
Another all Premier League tie pits new Aston Villa boss Tim Sherwood against local rivals West Brom.
Bradford have already beaten Premier League sides Chelsea and Sunderland to reach the quarter-finals and, as their chairman Mark Lawn hoped, they will take on Championship opposition in the form of Reading, winners over Derby County on Saturday.
The first tie out of the hat saw Liverpool given a home draw against Blackburn Rovers, who thrashed Stoke City 4-1 on Saturday thanks to a hat-trick from striker Josh King.
Quarter-final draw:
Liverpool v Blackburn Rovers
Bradford City v Reading
Manchester United v Arsenal
Aston Villa v West Brom
Ties to be played March 7 and 8.