Should United come through their tie against Reading on Monday, either Chelsea or Middlesbrough await.
The Blues beat Brentford 4-0 in a fourth-round replay on Sunday.
2011 winners Manchester City were drawn at home to Championship side Barnsley. City beat Leeds 4-0 on Sunday while Barnsley were 3-1 winners at MK Dons on Saturday.
Everton, who were pegged back in the final moments of their fifth-round match at Oldham on Saturday to draw 2-2, will host Wigan if they win the replay at Goodison.
The other last-eight clash will be between Millwall, 3-0 winners at Luton Town, and Blackburn Rovers, who shocked Arsenal 1-0 at the Emirates Stadium.
Oldham or Everton v Wigan Athletic
Manchester City v Barnsley
Manchester United or Reading v Middlesbrough or Chelsea
Millwall v Blackburn Rovers