Leeds, the League One leaders, scored the biggest Cup upset for years when they beat United 1-0.
NEWS: Leeds stun United at Old Trafford
Holders Chelsea, who beat Watford 5-0 on Sunday, will play another Championship side when they travel to Preston North End - their first visit to Deepdale for 29 years. Preston crushed Colchester United 7-0 in the third round.
Arsenal, who scored two late goals to beat West Ham United 2-1 at Upton Park on Sunday, were handed another tricky away tie at Premier League Stoke City, who saw off non-League York City 3-1 on Saturday.
There are three other potential all-Premier League matches.
Burnley will visit Liverpool if Rafa Benitez's men win their replay against Reading at Anfield, while Sunderland will face Portsmouth if Pompey win a replay at Coventry City.
Everton, who beat Carlisle United on Saturday, will host Premier League rivals Birmingham City if they get past in-form Championship side Nottingham Forest in a replay.
Manchester City, who under new coach Roberto Mancini are looking to win the FA Cup for the first time since 1969, should be too strong for Scunthorpe United, struggling at the wrong end of the Championship.
Leeds will look forward to their trip to London to face Spurs having become the first team to knock Manchester United out at the third round stage since Bournemouth - managed by Harry Redknapp - beat United in 1984.
By a twist of fate Redknapp is now manager of Tottenham - 4-0 winners over Peterborough United on Saturday - and will be determined not to lose to the team of Leeds manager Simon Grayson who has now emulated his achievement.
Grayson told Sky Sports: "It's another fantastic draw for us, against another top Premier League club and a fixture that would have been a Premier League game a few years ago. We are looking forward to it."
The Yorkshire club have only beaten Spurs once in the competition - in 1972 the year of their only FA Cup success.
Tottenham's former Leeds players Aaron Lennon, Robbie Keane and Jonathan Woodgate will be relishing their old club's visit.
The ties will be played on the weekend of January 23/24.
