The Millers face the unenviable task of trying to stop Pep Guardiola’s champions and current Premier League leaders on the first weekend of 2019.
Also tackling Premier League opposition in the Cup will be Barnsley, who travel to Burnley.
The South Yorkshire duo are among six White Rose sides on the road following a draw made by Ruud Gullit and Paul Ince.
Huddersfield Town, Yorkshire’s sole top flight representative, head to Bristol City, while Hull City and Leeds United are bound for the capital to tackle Millwall and QPR respectively.
Doncaster Rovers travel to Preston North End, while both Sheffield clubs have been drawn at home.
United will tackle Barnet of the National League at Bramall Lane, while Wednesday will host Luton Town.
An all-Yorkshire tie is also a possibility after the winners of next Tuesday’s replay between Bradford City and Peterborough United were handed a trip to Middlesbrough.
Solihull Moors were handed a dream third round draw against Premier League aristocrats Arsenal if they can get past Blackpool.
The National League outfit, who drew 0-0 with the League One Tangerines in the second round on Friday night, have been given the best possible incentive as they prepare for the replay.
National League North Southport too have an enticing prospect in line if they can get past League Two Tranmere with Tottenham drawn to face the winners.
Leicester, Premier League champions in 2016, will travel to either National League Wrexham or Newport of League Two, while Watford are also on the road at National League South Woking.
Everton were drawn to face Lincoln City.
FA Cup draw third round: Accrington v Ipswich, Aston Villa v Swansea, Bolton v Walsall/Sunderland, Bournemouth v Brighton, Brentford v Oxford, Bristol City v Huddersfield, Burnley v Barnsley, Chelsea v Nottm Forest, Crystal Palace v Grimsby, Derby v Southampton, Everton v Lincoln, Fulham v Oldham, Gillingham v Cardiff, Guiseley/Fleetwood v AFC Wimbledon, Man City v Rotherham, Man Utd v Reading, Middlesbrough v Peterborough/Bradford, Millwall v Hull, Newcastle v Blackburn, Norwich v Portsmouth, Preston v Doncaster, QPR v Leeds, Sheff Utd v Barnet, Sheff Wed v Luton, Shrewsbury v Stoke, Solihull/Blackpool v Arsenal, Tranmere/Southport v Tottenham West Brom v Wigan, West Ham v Birmingham, Woking v Watford, Wolves v Liverpool Wrexham/Newport v Leicester. Ties to be played January 4-7.