City secured their place in the last eight by beating Basel, having won the away leg 4-0, but then lost 2-1 at home in the second leg.
Liverpool saw off Porto in the round of 16, which also saw an impressive away performance with a 5-0 victory at the Estadio do Dragao ahead of a routine goalless draw at Anfield.
Anfield will host the first leg of the quarter-final tie, which will be played on April 3 or 4 with the second leg to be held on April 10 or 11.
Elsewhere in Friday’s quarter-final draw, holders Real Madrid were handed a tie against Italian giants Juventus, who beat Tottenham in the last round.
La Liga leaders Barcelona will play Roma, while the reward for Sevilla’s victory over Manchester United was a date against Bayern Munich of Germany.
The two English clubs have already played each other twice in the Premier League this season.
Manchester City romped to a 5-0 home win during September, with Gabriel Jesus and Leroy Sane both scoring twice after Sadio Mane had been sent off.
Liverpool, though, gained revenge at Anfield in January when they inflicted what so far has been the only Premier League defeat for City, who were beaten 4-3 with two late goals adding some respectability to the final scoreline.