Chelsea, the defending champions who are looking to win the competition for a fourth time in five years, secured their place in round five with a 4-0 fourth-round replay defeat of Brentford and will face npower Championship side Middlesbrough next.
The winner of that tie will then progress into the quarter-finals to meet runaway Barclays Premier League leaders United, who beat Reading 2-1 in the fifth-round of the competition.
Reigning Premier League champions Manchester City, who saw off Leeds 4-0 on Sunday in their fifth-round clash, were handed a home tie with Championship outfit Barnsley in the last eight as they look to repeat their 2010 FA Cup success.
Blackburn's reward for their shock 1-0 victory over Arsenal at the Emirates Stadium on Saturday was an away tie with fellow Championship side Millwall, who ended the run of non-league Luton in the fifth round with a 3-0 triumph.
The other quarter-final will see Wigan, 4-1 victors over Huddersfield on Sunday afternoon, travel to either Oldham or Everton.
League One Oldham, the lowest-ranked side still in the tournament, earned a fifth-round replay with Everton at Goodison Park after Matt Smith scored a stoppage-time equaliser in a 2-2 draw yesterday.
The quarter-final ties will be played on the weekend of March 9 and 10.
FA Cup Quarter-final Fixtures
Oldham/Everton vs Wigan
Man City vs Barnsley
Man United vs Middlesbrough/Chelsea
Millwall vs Blackburn.