With all big clubs avoiding each other, Arsenal host West Ham while Chelsea face Bournemouth at Stamford Bridge. The final tie will see Leicester host Premier League leaders .
The quarter-final ties are scheduled to be played on 19-20 December.
Reacting to the draw on his social media account, Bristol City head coach Lee Johnson extended an invitation to United manager Jose Mourinho for a post-match drink.
He wrote: 'Wecome to Bristol City Jose ... "pop in to the office for a drink after aye".
The draw, which was originally scheduled to take place at 4pm BST,following what was described as a 'minor technical glitch at Twitter HQ'.
A Twitter spokesman said: 'We're sorry for the issues that led to the delay of today's Round Five draw. We would also like to apologise to the thousands of supporters who had been patiently awaiting news of who their team would be playing.'
Manchester United, the defending champions of the trophy, made it into the hat for the quarter-finals after a solid 2-0 win over Swansea with Jesse Lingard scoring goals either side of half-time.
Their noisy neighbours and Premier League leaders Manchester City had it far from easy as they required a penalty shootout to down a brave Wolves team at the Etihad Stadium.
Similarly, Arsenal looked set to exit the competition to lower league opposition at home as Norwich outplayed a lethargic Gunners before youngster Eddie Nketiah came off the bench to score twice to send them through.
Chelsea needed several stunning saves from Willy Caballero to keep Everton at bay in their fourth round tie while West Ham completed a stunning comeback to knock out Tottenham and possibly save Slaven Bilic from the sack.
Bristol City are now the only non-Premier League club left in the draw after an impressive 4-1 win over Crystal Palace while Leicester and Bournemouth continued their pursuit of silverware with 3-1 wins over Leeds United and Middlesbrough respectively.
-Dailymail