The Blues secured a last-eight berth thanks to a 10-2 aggregate victory over Schalke in the previous round.
Spurs booked their place with a fine 4-0 aggregate victory over Borussia Dortmund.
City travel to Tottenham for the first leg on Tuesday 9 April, with the return game at the Etihad scheduled for Wednesday 17 April.
The winner will go on to face either Ajax or Juventus in the semi-final.
Full quarter-final draw:
Ajax v Juventus
Liverpool v Porto
Tottenham v CITY
Barcelona v Manchester United*
*United will play the first leg at home to avoid clashing with City's home fixture.
Tottenham or CITY v Ajax or Juventus
Barcelona or Manchester United v Liverpool or Porto
9/10 April: Quarter-finals, first leg
16/17 April: Quarter-finals, second leg
30 April/1 May: Semi-finals, first leg
7/8 May: Semi-finals, second leg
Saturday 1 June: Final – Estadio Metropolitano, Madrid