Crystal Palace, who came close to causing an FA Cup shock at home to Aston Villa on Sunday, will face either Reading or West Brom in the last eight if they come through a replay at Villa Park.
Portsmouth, who have been given additional time to get their finances in order following a High Court hearing this week, have been drawn against fellow Premier League side Birmingham at Fratton Park.
Pompey beat rivals Southampton in the fifth round on Saturday, while Birmingham are in the quarter-finals after overcoming Championship side Derby.
Finally Fulham, who beat Notts County 4-1 on Sunday, have been drawn at home in the last eight against either Bolton or Tottenham, who drew 1-1 at the Reebok Stadium also on Sunday.
Full quarter-final draw: Chelsea v Man City/Stoke City; Fulham v Bolton/Tottenham; Reading/West Brom v Crystal Palace/Aston Villa; Portsmouth v Birmingham City.
Ties to be played over the weekend of March 6.
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