The match fixtures are shown on an electronic panel after the draw of the quarterfinals of UEFA Champions League 2013/14 at the UEFA Headquarters in Nyon, Switzerland (Photo: AP)
Spanish clubs Barcelona and Atletico Madrid will also meet in the quarterfinals, while Paris Saint-German and Chelsea were drawn together.
The matches will be played on April 1-2 and 8-9.
Five-time champion Bayern travels to Manchester for the first leg, with the Bundesliga champions looking to heap more pressure on United manager David Moyes. United is seventh in the Premier League and likely needs to win the Champions League to qualify for next season's competition.
Dortmund lost to German rival Bayern in last year's final after eliminating Madrid, which hosts the first leg following an unbeaten run of 31 games in all competitions.
Three Spanish clubs reached the quarterfinals and at least one will make it to the semifinals. While Barcelona has had an irregular season, it has excelled in the Champions League but has yet to beat Atletico in three meetings this season.
Jose Mourinho will need to come up with a strategy to stop Zlatan Ibrahimovic when English leader Chelsea travels across the channel to face French champion PSG in the first leg.
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