Premier League leaders United will travel to Chelsea for the first leg and will have the home advantage at Old Trafford for the return leg. United beat Chelsea on penalties in the final in Moscow three years ago.
Holders Inter Milan have been drawn against German club Schalke 04 with Ukrainian club Shakhtar Donetsk facing a daunting trip to tournament favourites Barcelona for their first leg before hosting the Spaniards in the second leg.
The third English club Tottenham have been drawn against Spanish giants Real Madrid.
Draw for the Champions League quarter-finals:
Real Madrid (ESP) v Tottenham Hotspur (ENG)
Barcelona (ESP) v Shakhtar Donetsk (UKR)
Chelsea (ENG) v Manchester United (ENG)
Inter Milan (ITA) v Schalke 04 (GER)
Ties take place on April 5/6 and April 12/13