Alan Mahon's side made it through to the quarter-finals with a 10-0 win over Ipswich Town on Sunday afternoon, while the fifth-round tie between Leicester and Reading was postponed due to severe weather conditions.
City, the current holders, are looking to win the competition for the third time in four years.
Last-eight ties are scheduled for Sunday, 15 March, with the final at Wembley on 9 May.
Full quarter-final draw:
Crystal Palace or Brighton v Birmingham City
Bristol City or Everton v Chelsea
Arsenal or Lewes v Coventry United or Tottenham
Leicester or Reading v Manchester City