The Magpies beat West Bromwich Albion 3-2 at The Hawthorns on Tuesday night to reach the last eight of the competition for the first time since 2006.
And they will face City - who won 1-0 at Sheffield Wednesday - after the draw was made by former Magpies winger Chris Waddle and ex-Arsenal defender Martin Keown at Hillsborough on Wednesday evening.
Pep Guardiola's side won 6-0 against Watford in last year's final at Wembley to lift the trophy. Newcastle and City have already met in the league at St. James' Park - a thrilling 2-2 draw in November.
Quarter-final ties will be played between Friday, 20th March and Sunday, 22nd March, meaning United's Premier League home game against Aston Villa on Saturday, 21st March will be rescheduled.
It also means the Premier League match between City and the Magpies at the Etihad Stadium on Saturday, 18th April will be postponed due to one of the sides now being guaranteed a place in the semi-finals.
New dates for those fixtures, and ticket details for the FA Cup quarter-final, will be announced on nufc.co.uk in due course.
The full quarter-final draw is:Sheffield United v ArsenalNewcastle United v Manchester CityNorwich City or Tottenham Hotspur v Derby or Manchester UnitedLeicester City v Chelsea