United will travel to Bayern Munich for the first leg of their quarter-final in a rematch of the 1999 European Cup final, where the English Premier League side snatched two goals at the death for an extraordinary 2-1 win.
"This is a very difficult but also attractive draw," Bayern General Director Karl-Heinz Rummenigge told his club website.
"Manchester are a team that is up there at the very top. We will have to reach our limits and even exceed them if we want to progress. Manchester are the favourites and that is where it is attractive in that we can trip up the favourite."
Manchester United manager Sir Alex Ferguson told the club's website:
"The history of playing Bayern in past European ties tells you it's going to be a very difficult tie for us.
"It'll be a fantastic atmosphere in both games. They have a great stadium and a good pitch and we're up against a good experienced team.
"Hopefully we'll have everyone we need for the game and if we do then we have got a good chance."
The winners will face Olympique Lyon - conquerors of Real Madrid in the previous round and Liverpool in the group stage - or Girondins Bordeaux in the semi-finals after the two French clubs were pitted together in the last eight of the draw made at UEFA's headquarters in Nyon.
Inter Milan's reward for knocking out Chelsea is a quarter-final against CSKA Moscow, with either Barcelona or Arsenal waiting if they get past that hurdle.
Inter chief executive Ernesto Paolillo told Sky television: "This isn't an easy tie. It would be wrong to underestimate it for two reasons. First CSKA Moscow are at the start of their season (in Russia). So they are rested and fresher than any other team at the moment.
"Secondly, they play on a synthetic pitch, which favours less technical teams because players have less control of the ball.
Inter ambassador and former player Luis Figo told Sky: "You cannot pick and choose at this stage, but logically it's better if you manage to avoid sides like Barcelona and Manchester United."
Barcelona's match against Arsenal is a repeat of the 2006 final which the Spanish club won 2-1 in Paris. The London club will host the first leg.
Arsenal manager Arsene Wenger told the club website:
"I believe we will not be favourites but, for me, it will be a 50-50 game, that's how we have to take it.
"It will be an interesting, exciting game. We have to make sure we have the belief and focus right."
While Barca sporting director Txiki Begiristain told the Catalan club's website: "This will be the most spectacular of all the quarter-finals in terms of football. These are two teams that play open football and the match will be a duel to keep hold of the ball."
Club vice-president Rafael Yuste added: "If we get a good result in England the fans will be key for the second leg at the Nou Camp. We'll be able to count on a 12th player, the supporters."
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