Barcelona will take on familiar foes AC Milan with the first leg at the San Siro. Their most recent encounter was in the quarter-finals last season, when Barca prevailed 3-1 on aggregate with Lionel Messi scoring twice in the second leg.
German champions Borussia Dortmund face a tricky clash with Shakhtar Donetsk while local rivals Schalke take on Galatasaray.
Big-spending Ligue 1 outfit Paris St Germain will fancy their chances of beating Valencia, whose form has been patchy this season, and Malaga, who are performing better in the Primera Division, meet Portuguese champions Porto.
Cristiano Ronaldo scored 118 goals in 292 appearances for United, but it was another Ronaldo - the Brazilian striker - who took the honours when the clubs last faced each other in 2003, scoring a hat-trick at Old Trafford as Real went through 6-5 on aggregate.
The clubs first met in the European Cup in April 1957, when Real triumphed 5-3 in the semi-final. Ronaldo's last match for United was in the Champions League final defeat by Barcelona, after which he moved to Madrid for a world record £80million transfer fee.
Celtic boss Neil Lennon said: "Well in terms of glamour it's a beauty. Juventus are one of the great traditional European teams. They're current Italian champions and finished the group stage very strongly. So in terms of glamour it's great, in terms of qualification it is going to very tough, but it could have been any of the other teams and it would have been just as tough.
Bayern Munich chairman Karl-Heinz Rummenigge is optimistic his side can see off Arsenal. He told www.uefa.com: "I think we enter this match as the slight favourites, but we can be making the mistake of underestimating them.
"Arsenal have struggled at times in the league over the last few months, which a look upon the standings in the Premier League will tell you. But by no means will this be an easy game for us. I look forward to our game in London and back home in Munich."
United will play the first leg at the Bernabeu on February 13 with the return at Old Trafford on March 5.
Arsenal will play a week later with the first leg at the Emirates Stadium on February 19 and the second at the Allianz Arena in Munich on March 13.
Celtic will take on Juventus at Celtic Park on February 12, with the return leg in Turin on March 6.
Champions League last 16 draw in full
Galatasaray v Schalke
Celtic v Juventus
Arsenal v Bayern Munich
Shakhtar Donetsk v Borussia Dortmund
AC Milan v Barcelona
Real Madrid v Manchester United
Valencia v Paris Saint-Germain