First out of the draw were Ajax, who dismantled the reigning Champions at the Santiago Bernabéu.
The first legs will be played on April 9-10, with the return games on April 16-17.
Liverpool defender Virgil van Dijk heads home during their win over Bayern Munich.
Both Manchester clubs progressed to the last eight of the competition, but UEFA regulations do not allow two teams in the same city to play on the same day, nor on consecutive nights. The 31-year-old Barcelona forward is also the leading scorer in this season's competition with eight goals.
United's two-legged affair against Barcelona will be a rematch of both the 2009 and 2011 finals, which saw Sir Alex Ferguson's men defeated by Pep Guardiola's brilliant Catalan outfit on both ocassions.
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Not all the memories are bad for Man United, however.
The way the draw works United would play Liverpool if they made it through to the semi-finals and City would play Juventus.
Winning the FA Cup is "not really on" Manchester City's radar with fans and players split on what is their priority, Danny Mills says. However, Porto have the second leg at home and if they can avoid a heavy defeat at Anfield in the first game, they will fancy their chances of turning it around. It's a tough side but any other team in the Champions League would be the same.
Sadio Mane scored a hat-trick at the Estadio do Dragao in a 5-0 win before completing the task at home by drawing 0-0.
"The result in the end was a odd one; it was very good for us, of course, but it was a unusual one". The boost of having the first leg being played at Spurs' new stadium could be crucial. Ajax won in 1973, the third of its four European titles, and Juventus won its second title in 1996.
And Juventus' meeting with Ajax will be a rematch of the 1996 Champions League final at Rome's Stadio OIimpico, a match that the Italians won via a penalty shootout. The first showing was in 2001/02 when the Merseyside club lost 4-3 on aggregate to Bayer Leverkusen who would go on to finish runners-up to Real Madrid.