United must get past last season's winners over two legs next month to keep alive their chances of winning a trophy in David Moyes' first season.
Chelsea will face PSG for a place in the last four, while Real Madrid play Borussia Dortmund in a repeat of last season's semi-final and Barcelona are drawn with La Liga rivals Atletico Madrid.
United and Bayern have plenty of recent history in the competition: the Germans were the opposition when United completed the treble in 1999, while Bayern were victorious at the quarter-final stage in 2001 and 2010.
But Pep Guardiola's Bayern have been in amazing form this season, unbeaten in the Bundesliga and having won every match at home in that competition.
David Moyes can take some solace from the fact that the only two clubs who have prevented Bayern from winning at home this season were both English: Manchester City actually won 3-2 at the Allianz Arena while Arsenal achieved a 1-1 draw in the last round.
Chelsea's draw against PSG will see Mourinho come up against his former striker Zlatan Ibrahimovic, whom he managed at Inter Milan before the Swede was sold to Barcelona.
Ibrahimovic has spoken of his admiration for Mourinho, saying in his autobiography that he "was basically willing to die for'' the Portuguese boss.
Barcelona v Atletico Madrid
Real Madrid v Borussia Dortmund
Paris St Germain v Chelsea
Manchester United v Bayern Munich
Ties to be played on April 1/2 and 8/9