United made it through to the last eight on Wednesday after they overturned a two-goal deficit on a thrilling night against Olympiacos.
Three-time winners United are 18-1 shots to lift the trophy, but Wednesday's win over Olympiacos proved they are more than capable of upsetting the odds against a strong field.
As well as their Premier League rivals, the quarter-final draw contains Bundesliga giants Bayern Munich and Borussia Dortmund, Spain's title-chasing trio Barcelona, Real Madrid and Atletico Madrid, and big-spending French champions Paris St Germain.
The dark clouds that hung over David Moyes have lifted - for the time being at least - and there is a sense of optimism around the club about their European campaign.
England striker Wayne Rooney said: "I think in cup football anything can happen. With a bit of luck hopefully we can go far.
"We're in the quarter-final so you never know what can happen from there.
"You look at the three Spanish teams left in the competition, they're all good teams. Chelsea are probably playing the best football this season and obviously you've got Bayern Munich.
"PSG are playing well. It's a talented group of teams so whoever we get I'm sure it will be a tough game.
Chelsea's Premier League campaign faltered at the weekend when they lost to Aston Villa, but they bounced back with a 2-0 win over Galatasaray on Tuesday.
Having kept Didier Drogba quiet throughout the 3-1 aggregate win, the Blues' confidence is high ahead of the draw, which takes place at 1100 GMT.
"Didier is one of the best players in the world so the result showed we are playing well," said defender Gary Cahill, who scored the second goal in Chelsea's 2-0 return-leg win.
"The opponents in the next round will be top drawer but we are playing well and if we play like we did (in the second leg) we will be okay."
Manager Jose Mourinho's own hunger for success in the competition cannot be doubted either. If Chelsea win the Champions League, he will be the only manager other than Bob Paisley to lift the trophy three times.
Despite all the bravado from both camps, perhaps the best way for England to have a team in the last four is if United and Chelsea draw each other in the quarter-finals.
The other six teams left in the hat have won 19 European Cups between them and those that have not won the competition are certainly capable of doing so this time around.
Bayern coach Pep Guardiola, who won the Champions League twice with Barcelona, has some of the best players in the world at his disposal.
The Germans, who are unbeaten at the top of the Bundesliga, are favourites for the trophy, just ahead of Real Madrid and Barca.
United or Chelsea would have trouble shackling Real star Cristiano Ronaldo, who needs two goals to break Lionel Messi's record of 14 in one Champions League campaign.
Messi played his own part in helping Barca to a 4-1 win over Manchester City in the previous round.
Barcelona look set to be pushed into third place in the Spanish Primera Division by Atletico, who underlined their status as Champions League challengers by beating AC Milan 5-1 in the second round.
Last year's beaten finalists Dortmund still have some of the most exciting players on the continent, although one of them - Robert Lewandowski - is suspended for the first leg of the quarter-finals.
And with hundreds of millions of pounds' worth of talent at their disposal, PSG would also be formidable opponents.
The first legs of the quarter-finals will take place on April 1 and 2, and the return fixtures will be played the following week as teams bid to reach May 24's final in Lisbon.