Holders Chelsea have been drawn to face 12-time winners Manchester United in the FA Cup fifth round, a repeat of last year's final.
With Arsenal, Tottenham Hotspur and Liverpool already out of the competition, today's draw means that, at most, only two of the top six will progress to the quarter-finals.
Chelsea won last year's final at Wembley 1-0 thanks to an Eden Hazard penalty.
Giant-killers Newport County, who play in League Two (fourth tier), could face Premier League champions Manchester City if they can win their replay against Middlesbrough.
Newport, who have already knocked out top-flight Leicester City, drew 1-1 at Middlesbrough at the weekend and host the replay on February 5th.
Doncaster Rovers, who play in League One (third tier) and are in the fifth round for the first time in 63 years, will host Premier League Crystal Palace.
League One AFC Wimbledon, who knocked out top-flight West Ham United, take on Championship (second-tier) Millwall, who eliminated Everton in the fourth round.
The fifth-round ties will take place between February 15-18.
Bristol City v Shrewsbury Town or Wolverhampton Wanderers
AFC Wimbledon v Millwall
Doncaster Rovers v Crystal Palace
Middlesbrough or Newport County v Manchester City
Chelsea v Manchester United
Swansea v Barnet or Brentford
Portsmouth or Queens Park Rangers v Watford
Brighton & Hove Albion or West Bromwich Albion v Derby County