Arsene Wenger's side have not met Tottenham in the competition since the 2001 semi-final - a game they won 2-1 at Old Trafford to edge 3-2 ahead in FA Cup meetings between the sides.
January's match at the Emirates Stadium will be one of five all-Premier League ties.
Manchester United face Swansea City, West Brom have been drawn against Crystal Palace, Newcastle United entertain Cardiff City and Fulham travel to Norwich City.
Holders Wigan Athletic will begin their defence at home to League One side MK Dons, while beaten 2013 finalists Manchester City head to Ewood Park to face Blackburn Rovers.
Seven-time winners Liverpool could face Oldham Athletic for the third successive year if Lee Johnson's side can overcome non-league Mansfield Town in a second-round replay.
Oldham knocked Brendan Rodgers' side out of the competition at the fourth-round stage last season, with Liverpool winning 5-1 in January 2012 at Anfield.
Elsewhere, Chelsea travel to in-form Championship side Derby County, while Southampton face Burnley, the leaders in England's second tier, at St Mary's Stadium.
The three non-league sides definitely in round three - Kidderminster Harriers, Grimsby Town and Macclesfield Town - have respective home draws against Peterborough United, Huddersfield Town and Sheffield Wednesday, while Wrexham could face Charlton Athletic if they get past Oxford United.
All ties are set to take place on the weekend commencing January 4.
FA Cup third round draw in full: Barnsley v Hartlepool or Coventry City; Yeovil Town v Leyton Orient; Liverpool v Oldham Athletic or Mansfield Town; Nottingham Forest v West Ham United; Bristol City v Watford; Southend United v Millwall; Middlesbrough v Hull City; West Brom v Crystal Palace; Kidderminster v Peterborough United; Doncaster Rovers v Stevenage; Stoke City v Leicester City; Southampton v Burnley; Newcastle United v Cardiff City; Rochdale v Leeds United; Wigan Athletic v MK Dons; Charlton Athletic v Wrexham or Oxford United; Manchester United v Swansea City; Port Vale v Plymouth Argyle; Norwich City v Fulham; Aston Villa v Sheffield United; Macclesfield Town v Sheffield Wednesday; Sunderland v Carlisle United; Bolton Wanderers v Blackpool; Blackburn Rovers v Manchester City; Everton v QPR; Brighton and Hove Albion v Reading; Arsenal v Tottenham; Birmingham City v Bristol Rovers or Crawley Town; Grimsby Town v Huddersfield Town; Ipswich Town v Preston North End; Derby County v Chelsea; Bournemouth v Fleetwood Town or Burton Albion.
