In the stand out tie of the round, Maurizio Sarri’s side will take on Ole Gunnar Solskaer’s improving Red Devils at Stamford Bridge next month.
United have won all eight of Solskaer’s matches in charge since the Norwegian succeeded Jose Mourinho last month, including an impressive 3-1 victory against Arsenal at Emirates Stadium in the fourth round on Friday.
Eight-time winners Chelsea reached round five courtesy of a comfortable 3-0 victory over Sheffield Wednesday on Sunday.
Eden Hazard scored the only goal from the penalty spot when the sides met at Wembley in May, with Chelsea lifting the trophy for the fifth time in 11 years.
League One's Doncaster, through to the fifth round for the first time in 63 years, will host Crystal Palace.
AFC Wimbledon, who beat West Ham in round four, have been handed a home draw against Millwall, who also overcame Premier League opposition in the form of Everton.
Swansea will play either Barnet or Brentford, who face each other at the Hive on Monday evening.
The FA Cup fifth round ties will take place the weekend of 15-18 February.
FA Cup fifth round draw in full:
- Bristol City v Shrewsbury Town or Wolverhampton Wanderers
- AFC Wimbledon v Millwall
- Doncaster Rovers v Crystal Palace
- Middlesbrough or Newport County v Manchester City
- Chelsea v Manchester United
- Swansea v Barnet or Brentford
- Portsmouth or Queens Park Rangers v Watford
- Brighton or West Brom or Derby County