The tie at Old Trafford is the reward for Jose Mourinho's men, who beat Huddersfield Town 2-0 in Saturday's fifth-round tie at the John Smith's Stadium thanks to a brace from Romelu Lukaku.
Swansea City could host top-flight rivals Tottenham, who face Rochdale on Sunday, but the Welsh club must first see off Sheffield Wednesday at the second attempt after their initial meeting ended in a 0-0 stalemate.
The four quarter-finals are scheduled to take place on the weekend of Saturday 17th and Sunday 18th March.
There will be no replays in the quarter-finals, following a change to the rules last season.
Of the ten clubs remaining in the competition, seven are from the Premier League.
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City's cross-town rivals Man Utd will be looking to take a step closer to the final when they take on Brighton, who have never won the FA Cup.
When is the semi-finals and final?However, they needed a replay to beat both sides.
Pep Guardiola's Manchester City will enter the FA Cup quarter-final aware that their midweek thrashing of Basel in the Champions League keeps the improbable dream of a quadruple alive.
The Latics' reward is a home clash against Southampton, and the League One side are just one victory away from an unlikely spot in the last four.