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 on Sunday and will face npower Championship side Middlesbrough next.
The winner of that tie will then play either runaway Barclays Premier League leaders United or Reading in the quarter-finals, with those teams playing their fifth-round clash at Old Trafford on Monday night.
Reigning Premier League champions Manchester City, who saw off Leeds 4-0 in their fifth-round clash, were handed a home tie with Championship outfit Barnsley in the last eight as they look to repeat their 2011 FA Cup success.
Blackburn's reward for their shock 1-0 victory over Arsenal at Emirates Stadium 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, 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.
FA Cup quarter-final draw:
Oldham or Everton v Wigan Athletic
Manchester City v Barnsley
Manchester United or Reading v Middlesbrough or Chelsea
Millwall v Blackburn
Ties to be played on March 9/10.