Saints have been drawn away to Wigan or Premier League leaders Manchester City in the FA Cup quarter-finals.
The sides meet in their fifth round tie on Monday night.
City are massive favourites to get through that match and would also be a huge test for Saints, albeit Mauricio Pellegrino's side were only beaten by a late goal in their Premier League visit to the Etihad Stadium earlier in the season.
The game will be played between March 16-19, meaning Saints' trip to Swansea will have to be rearranged.
The full quarter-final draw is:
Sheffield Weds/Swansea v Rochdale/Tottenham
Man Utd v Brighton
Leicester City v Chelsea
Wigan/Man City v Southampton
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