The draw saw Arsenal face the side who eliminated them from the competition last season in Olympiakos, while AS Roma and Shakhtar Donetsk -- Champions League regulars in recent seasons -- will play one another.
Between them, United and Milan have won a total of 10 European Cup/Champions League titles. They last faced each other in the Champions League round of 16 in 2010, when United won 7-2 on aggregate.
Should Unai Emery lead Villarreal to success in this competition, he would become the first manager in men's football to win a European competition four times, having already won three Europa League titles with Sevilla.
Tottenham's tie with Dinamo Zagreb is expected to be reversed by UEFA to avoid a clash with fellow London side Arsenal playing at home on the same night.
The draw was completely open, with every team able to face any possible opponent. The ties will be played over two legs, despite the latter stages of last season's campaign having been a single tie, with the first legs on March 11 and the returns a week later on March 18.
UEFA will draw the quarterfinal and semifinal ties on the same day this season, scheduled for March 19.
The final will take place in Poland at the Stadion Miejski in Gdansk on Wednesday, May 26.
ROUND OF 16
Ajax vs. Young Boys
Dynamo Kiev vs. Villarreal
AS Roma vs. Shakhtar Donetsk
Olympiakos vs. Arsenal
Dinamo Zagreb vs. Tottenham Hotspur
Manchester United vs. AC Milan
Slavia Prague vs. Rangers
Granada CF vs. Molde FK