The current Premier League leaders beat Fulham on the way to their last FA Cup triumph in 2003-04 but they have not met Swansea since the 1982-83 season.
Sunday's other winners Everton face Premier League opposition again in the shape of West Ham or Middlesbrough while Chelsea - winners in 2007 - must travel to face the winners of the replay between Blackburn and Coventry.
The Sky Blues were just moments away from winning at Ewood Park yesterday but Christopher Samba's injury-time goal saw Rovers claim a 2-2 draw.
Cardiff or Arsenal, who have yet to play their fourth round clash, will play the Burnley for the right to face Sheffield United or Hull in the last eight.
FA Cup Sixth Round draw:
Blackburn or Coventry v Chelsea
Swansea or Fulham v Manchester United
Cardiff/Arsenal or Burnley v Sheffield United or Hull
Everton v West Ham or Middlesbrough
Ties to be played on the weekend of March 7-8.
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