Tony Adams' men face a tricky replay at Ashton Gate, but if they overcome the Robins they will play either Histon or Swansea at Fratton Park.
Elsewhere, the fourth round will feature a Merseyside derby after Liverpool and Everton were the first two balls drawn out of the hat.
Anfield will host the standout tie, a week after the pair meet at the same ground in the league, while other highlights include non-league Kettering entertaining Fulham and Arsenal travelling to Cardiff.
Nottingham Forest's reward for beating Manchester City is a trip to bitter rivals Derby while Coca-Cola League One Hartlepool, who beat Premier League Stoke, host West Ham.
Torquay, the other non-league side left in the draw, were also drawn at home with Coventry the visitors.
The ties will be played on the weekend of January 24/25.
Full draw:
– Birmingham/Wolves v – Middlesbrough
– Cardiff v Arsenal
– Chelsea/Southend v Ipswich
– Cheltenham/Doncaster v Aston Villa
– Derby v Nottm Forest
– Hartlepool v West Ham
– Hull/Newcastle v Millwall/Crewe
– Kettering v Fulham
– Leyton Orient/Sheff Utd v Charlton/Norwich
– Liverpool v Everton
– POMPEY/Bristol City v Histon/Swansea
– Southampton/Man Utd v Tottenham
– Sunderland v Blyth/Blackburn
– Torquay v Coventry
– Watford v Leicester/Crystal Palace
– West Brom/Peterborough v QPR/Burnley
Have a comment on the draw? Click here to leave a message and we will publish the best ones in The News on Monday.Never miss a Portsmouth FC story... Get them sent straight to your e-mail inbox.... click here to subscribe.
The full article contains 280 words and appears in The News newspaper.