The first leg will be played in Rome on 1st 0r 2nd April with the second leg taking place at Old Trafford on either 8th or 9th April.
Should they overcome the Italian side then United would play either Schalke or Barcelona in the semi-finals.
Full quarter-final draw:
Arsenal v Liverpool
AS Roma v Man Utd
Schalke 04 v FC Barcelona
Fenerbahce v Chelsea
Semi-final draw:
Arsenal/Liverpool v Fenerbahce/Chelsea
Schalke/Barcelona v AS Roma/Man Utd