Sheffield Wednesday will play Stevenage or Reading in their fourth-round tie if they beat Carlisle in their third-round replay, while United will face Championship rivals Preston at Bramall Lane.
United booked their place in the fourth round by beating Ipswich Town 1-0 last weekend, thanks to Nathan Thomas' stunner, but Wednesday must overcome Carlisle in a replay after their 0-0 draw at Brunton Park. The sides meet again a week tomorrow, at Hillsborough.
Liverpool will face West Brom, Yeovil will host Manchester United and Tottenham Hotspur will travel to Newport County.
Fourth-round ties will be played on the weekend of January 26-29.
Full draw: Liverpool v West Brom, Peterborough v Fleetwood or Leicester, Huddersfield v Birmingham, Notts County v Wolves or Swansea, Yeovil v Manchester United, Carlisle or Sheffield Weds v Stevenage or Reading, Cardiff or Mansfield v Manchester City, MK Dons v Coventry, Millwall v Rochdale, Southampton v Watford, Middlesbrough v Brighton or Crystal Palace, Bournemouth or Wigan v Shrewsbury or West Ham, Hull v Nottingham Forest, Newport County v Tottenham, Norwich or Chelsea v Newcastle, Sheffield United v Preston.