Sheffield United will host Arsenal, while Chelsea travel to Leicester.
The other last-eight tie will be Norwich taking on one of Derby or Manchester United, who play on Thursday night.
The Canaries booked their spot in the quarter-finals as Erik Lamela, Troy Parrott and Gedson Fernandes all missed from the spot for Tottenham in the penalty shoot-out.
FA Cup quarter-final draw in full:
- Sheffield United v Arsenal
- Newcastle v Manchester City
- Norwich v Derby/Manchester United
- Leicester v Chelsea
