Roma reached the last four of the competition - for the first time since it was rebranded - thanks to a 1-1 draw with Ajax on Thursday evening, enough for a 3-2 aggregate victory.
United, meanwhile, eased past Spanish side Granada to reach this stage.
The Premier League side will host the first leg of the semi-final at Old Trafford on Thursday 29 April, with the return leg scheduled for a week later on May 6 at the Stadio Olimpico.
Both games get underway at 21:00 CEST.
The Giallorossi defeated Sporting Braga, Shakhtar Donetsk and Ajax in the knockout stages of the competition.
United, meanwhile, have beaten Real Sociedad, AC Milan and Granada.
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