Eddie Howe's side saw off Blackburn Rovers in the fifth round on Tuesday night, winning 4-3 on penalties after Anthony Gordon and Sammie Szmodics exchanged goals in normal time at Ewood Park.
Goalkeeper Martin Dúbravka's heroics in the shoot-out - in which he saved from Szmodics and Dominic Hyam - sent United through and their reward is a trip to face the current FA Cup holders at the Etihad Stadium.
The Magpies - who have also beaten Sunderland and Fulham on their way to the last eight - will play Pep Guardiola's City on the weekend of Saturday, 16th March, meaning their Premier League game at Crystal Palace will be rescheduled.
United have already beaten City in the cup this year - Alexander Isak's goal earning the Magpies a 1-0 win in the Carabao Cup third round at St. James' Park earlier this season - but their trip to the Etihad will be their sixth away tie in a row across both domestic cup competitions.
A confirmed date for the quarter-final clash, and any further details about the tie, will be confirmed in due course.