The Toffees will face either a Merseyside derby or a trip to north London in the next round, while Manchester City's quadruple hunt sees them drawn against holders WiganRoberto Martinez will need to traverse an away trip to Arsenal in the FA Cup quarter-finals if he is to win the competition for a second consecutive season.
Everton, who beat Swansea City 3-1 at Goodison Park on Sunday to book their place in the sixth round, claimed a 1-1 draw at the Emirates Stadium earlier this season, but were hammered 4-0 by their Merseyside rivals at Anfield.
Elsewhere, Gus Poyet could be set for quick return to the Amex Stadium and former employers Brighton, with Sunderland drawn against either the Seagulls or Steve Bruce's Hull City.
Quadruple-chasing Manchester City, meanwhile, will face reigning FA Cup holders Wigan Athletic at the Etihad Stadium, with Manuel Pellegrini's men having defeated Chelsea 2-0 en route to the quarters.
The draw in full:
Arsenal vs Everton
Brighton or Hull City vs Sunderland
Sheffield United vs Sheffield Wednesday or Charlton
Manchester City vs Wigan Athletic