Stevenage, struggling in the lower half of the GM Vauxhall Conference this season, are likely to switch the tie to St James' Park, but their manager, Paul Fairclough, said: "We'll leave that up to the powers that be. We've got the draw we wanted."
Stevenage are 5,000-1 outsiders for the Cup, while Manchester United have been installed as 3-1 favourites after being drawn at home to either Peterborough or Walsall.
Middlesbrough's Paul Merson could find himself playing against his old club, Arsenal, if both clubs win their replays against Queen's Park Rangers and Port Vale respectively.
Coventry, having knocked out Liverpool, come up against further Premiership opposition in Derby County at Highfield Road while West Ham, narrow 2- 1 winners against non-League Emley, travel to Maine Road to face struggling Manchester City.
FA Cup fourth-round draw
AFC Bournemouth or Huddersfield Town v Wimbledon or Wrexham
Manchester City v West Ham Utd
Charlton Athletic v Darlington or Wolves
Stevenage Borough v Newcastle Utd
Tottenham Hotspur or Fulham v Barnsley
Queen's Park Rangers or Middlesbrough v Arsenal or Port Vale
Coventry City v Derby County
Leeds Utd v Grimsby Town
Watford or Sheffield Wednesday v Blackburn Rovers
Crystal Palace v Leicester City
Birmingham City v Stockport County
Manchester Utd v Peterborough Utd or Walsall
Bristol Rovers or Ipswich Town v Sheffield Utd or Bury
Cardiff City v Cheltenham Town or Reading
Portsmouth or Aston Villa v West Bromwich or Stoke
Hereford Utd or Tranmere Rovers v Sunderland
Ties to be played weekend of 24/25/26 January