Atletico Madrid take on Juventus, Tottenham Hostpur are up against Borussia Dortmund and Real Madrid face Ajax.
Barcelona play Lyon, Manchester City meet Schalke, while AS Roma vs Porto completes the round of 16 ties.
PSG, who finished ahead of Liverpool and Napoli to top Group C, are undefeated in Ligue 1 this season and will travel to Old Trafford before hosting United in Paris.
United are struggling in the Premier League this season and fell 19 points behind leaders Liverpool following their 3-1 defeat at Anfield on Sunday.
Bayern Munich and Liverpool, who are both five-times winners of the competition, will meet in a competitive game for the first time since 2001, while Spurs face Dortmund and City play Schalke in the other clashes between English and German opposition.
Juventus will face Atletico Madrid for the first time in the knockout phase of a European competition, as Cristiano Ronaldo returns to Madrid for the first time since leaving for Serie A. The Portuguese forward scored 22 goals against Atletico during his time at Real.
Schalke vs. Manchester City
Atletico Madrid vs. Juventus
Manchester United vs. Paris Saint-Germain
Tottenham vs. Borussia Dortmund
Lyon vs. Barcelona
Roma vs. Porto
Ajax vs. Real Madrid
Liverpool vs. Bayern Munich
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