Spanish clubs Barcelona and Atletico Madrid will also meet in the quarterfinals, while Paris Saint-German and Chelsea were drawn together.
The matches are April 1-2 and 8-9.
For Manchester United, an already arduous season just got tougher, with the English champions drawn Friday against the favorite and five-time champion.
''Perhaps they are not in the best form, but they are still a top team,'' said Bayern winger Arjen Robben, who scored a late goal to eliminate United from the competition at the same stage four years ago. ''It's a good draw, we play away first.''
United, under manager David Moyes, is seventh in the Premier League and likely needs to win the Champions League to qualify for next season's competition.
Bayern, meanwhile, is unbeaten in 50 Bundesliga games under coach Pep Guardiola and leads second-place Dortmund by 23 points.
Dortmund beat Madrid in the semifinals last year before losing to Bayern in the final at Wembley Stadium. But Madrid is unbeaten in 31 games in all competitions, and Cristiano Ronaldo leads the Champions League with 13 goals this season.
Three Spanish clubs reached the quarterfinals and at least one is guaranteed to play in the semifinals.
Although Barcelona has excelled in the Champions League, the Spanish champions have had an irregular season at home and have yet to beat Atletico in three games - all draws.
For Chelsea, coach Jose Mourinho will need to come up with a strategy to stop Zlatan Ibrahimovic when the English leaders travel across the channel to face French champion PSG in the first leg.
Mourinho led Chelsea against PSG in the tournament's group stage in 2004-05, when the Blues won 3-0 in Paris before a scoreless draw at home.