Premier League leaders United, who beat the Blues on penalties in Moscow three years ago, will enjoy home advantage in the second leg against the defending English league champions.
Tottenham were handed a glamorous but daunting clash against Real Madrid. However, they will take on the Spanish giants at White Hart Lane in the second match.
Spot of bother: Chelsea's John Terry missed his penalty as United won the 2008 final after a shoot-out
Quarter-final drawReal Madrid v Tottenham
Chelsea v Manchester United
Barcelona v Shakhtar Donetsk
Inter Milan v Schalke
First leg April 5/6
Second leg April 12/13
Favourites Barcelona, who conquered Arsenal in the last round, were handed one of the two supposedly easier draws and will take on Ukrainians Shakhtar Donetsk while defending champions Inter Milan will face German side Schalke 04.
The semi-finals were also drawn and the winner of the Chelsea-United clash will meet Inter or Schalke in the last four, with the winning English side at home second.
Should Spurs come through against Madrid, they will be at home first against Barca or Shakhtar.
While Chelsea are reigning champions in England, rivals United currently lead them by nine points in the Barclays Premier League and have history on their side in the competition.
The Blues have never won the Champions League, despite owner Roman Abramovich's best efforts.
The closest they came was in Russia in 2008 when United edged them over 120 minutes and penalties, having drawn 1-1 in normal time at the Luzhniki Stadium.
Spurs, winners against AC Milan in the last 16, face a tough task on paper against the record nine-time winners Real Madrid.
Just champion: Jose Mourinho and Inter Milan lifted European club football's biggest prize in Madrid last May
Semi-final drawInter/Schalke v Chelsea/Man U
Real/Spurs v Barca/Shakhtar
First leg April 26/27
Second leg May 3/4
However, Jose Mourinho's Spanish aristocrats have not picked up the trophy since 2002 and are regarded as some way short of their rivals Barca on current form.
The quarter-final first legs will be played on April 5 and 6 and the return matches the following week on April 12 and 13.
The semi-final first legs take place on April 26 and 27 with the second legs on May 3 and 4, with the final at Wembley on Saturday, May 28.
Should they beat Schalke, Inter would represent a tough test in the last four for Chelsea or United.
While they are without Mourinho, who led them to glory in the tournament last season, they are back in form since Leonardo replaced Rafa Benitez recently.
Schalke this week sacked Felix Magath despite his significant achievement in guiding them to this stage, appointing former Hoffenheim boss Ralf Rangnick as his successor.
Shakhtar are perennial Champions League stalwarts and are not to be taken lightly, especially as favourites Barca have injury problems as well as a Spanish league title race and domestic cup final to contend with.
Italian job: Peter Crouch was the Spurs hero with the only goal against AC Milan over two legs in the last round
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Explore more:People: Roman Abramovich, Jose Mourinho, Felix Magath, Rafa Benitez Places: Madrid, Moscow, United Kingdom, Russia