Swindon are hoping for a fourth-round tie against Manchester United after defeating Premier League Wigan 2-1 at the County Ground on Saturday.
After his side's win, Town chairman Jeremy Wray said: "Next round, let's hope it is Manchester away."
Non-league Wrexham remain in the hat, as do League Two sides Macclesfield, Crawley Town and Dagenham & Redbridge.
Wigan and Blackburn, who lost to Newcastle, were the only Premier League teams knocked out on Saturday.
Manchester United are in the draw after defeating rivals City 3-2, while Chelsea host Portsmouth, Sunderland travel to Peterborough and Sheffield Wednesday entertain West Ham in Sunday's other fixtures.
On Monday, Arsenal make the trip to Leeds.
Also still in the draw are Stevenage and Notts County, who beat Reading and Doncaster respectively, and MK Dons, who held QPR.
Swindon were the chief headline-makers, however, and manager Paolo di Canio - himself no stranger to the magic of the cup - was particularly proud of his players.
"My lads today deserve to have their names put on this stadium," he said.
"I know you normally do this when you win something important and I don't want a big statue but maybe a plaque. Today we did something special."
FA Cup fourth round draw:
2. Nottingham Forest or Leicester City.
3. Manchester United.
4. Dagenham & Redbridge or Millwall.
5. Crawley Town.
6. Notts County.
7. Aston Villa.
8. Tottenham Hotspur.
9. Sheffield Wednesday or West Ham United.
10. Milton Keynes Dons or Queens Park Rangers.
11. Hull City.
13. Brighton & Hove Albion or Wrexham.
15. Birmingham City or Wolverhampton Wanderers.
16. Norwich City.
17. Arsenal or Leeds United.
18. Derby County.
20. Swindon Town.
21. Swansea City.
22. Macclesfield Town or Bolton Wanderers.
23. Newcastle United.
25. Sheffield United.
27. Stoke City.
28. Chelsea or Portsmouth.
30. Peterborough United or Sunderland.
31. West Bromwich Albion.