Holders Manchester City will play Copenhagen in the last 16 of the Champions League, while Arsenal face Porto.
Danish champions Copenhagen, who finished above Manchester United in their group, reached the knockout stage for the first time since 2011.
The first legs take place on 13-14 or 20-21 February and the second on 5-6 or 12-13 March.
City and Arsenal are seeded, meaning they will play the second leg at home.
Record 14-time winners Real Madrid meet RB Leipzig, while Inter Milan – runners-up last season – play Atletico Madrid, and Bayern Munich face Lazio.
The final takes place at Wembley on 1 June.
Last-16 draw
Porto v Arsenal
Napoli v Barcelona
Paris St-Germain v Real Sociedad
Inter Milan v Atletico Madrid
PSV Eindhoven v Borussia Dortmund
Lazio v Bayern Munich
Copenhagen v Manchester City
RB Leipzig v Real Madrid
Premier League leaders Arsenal, who topped Group B but have never won the Champions League, beat Porto 5-0 at Emirates Stadium the last time they met in 2010.
Runners-up in Portugal’s Primeira Liga last season, Porto won four of their six Group H games to finish level on points with Group H winners Barcelona.
City won Group G with a 100% record, while Copenhagen, who lost 1-0 at United before beating them 4-3 at home, were runners-up to Bayern in Group A.
City have progressed from the last 16 in the past six seasons.
Analysis
Ian Dennis, BBC Radio 5 Live senior football reporter
The draw has been very kind to the two English teams and both should be very happy.
Arsenal have won all three previous encounters against Porto, although they have never won at the Estadio do Dragao.
City played Danish champions Copenhagen in the group stage last season, drawing 0-0 in Denmark but winning handsomely 5-0 at home.
Europa League play-off draw
The Europa League knockout round play-off draw also took place on Monday.
There was no British interest, with West Ham, Brighton, Rangers and Liverpool among the eight group winners automatically advancing to the last-16 stage, which will be played on 7 and 14 March.
They will be joined by the winners of the play-offs, contested by eight runners-up from the Europa League group stage and eight third-placed teams from the Champions League group stage.
These ties will be played on 15 and 22 February.
Feyenoord v Roma
Lens v Freiburg
Benfica v Toulouse
Galatasaray v Sparta Prague
AC Milan v Rennes
Young Boys v Sorting Lisbon
Sporting Braga v Qarabag
Shakhtar Donetsk v Marseille
Europa Conference League play-off draw
Eight teams are safely through to the last 16 of the Europa Conference League, including Aston Villa.
They will be joined by the winners of the knockout round play-offs, contested between eight runners-up from the Europa Conference League group stage and eight third-placed teams from the Europa League group stage.
These ties will also be played on 15 and 22 February.
Sturm Graz v Slovan Bratislava
Servette v Ludogorets
Union Saint-Gilloise v Eintracht Frankfurt
Real Betis v Dinamo Zagreb
Olympiakos v Ferencvarosi
Ajax v Bodo/Glimt
Molde v Legia Warsaw
Maccabi Haifa v Gent