While several of their Premier League rivals floundered in Saturday's third-round ties, Chelsea thrashed Southampton 5-1 and now face a trip to either fourth-tier Southend United or third-division Brentford.
City, 3-0 victors over Watford, will face either Premier League rivals Stoke City or Championship side Crystal Palace.
United needed an injury-time Robin van Persie equaliser to earn a 2-2 draw at West Ham United, with either Fulham or Blackpool awaiting the winners of the replay in round four.
Second-tier Brighton and Hove Albion's reward for eliminating Newcastle United, meanwhile, is a tie with either Swansea City or Arsenal, who drew 2-2 at the Liberty Stadium on Sunday.
Liverpool, 2-1 victors against non-league Mansfield Town on Sunday, will visit third-tier Oldham Athletic, who reached the semi-finals in 1994.
Fifth-division Macclesfield Town, conquerors of Championship leaders Cardiff City, could face Premier League opposition in the fourth round if Wigan Athletic come through their replay against League One Bournemouth.
Fellow Conference National representatives Luton Town will definitely face a side from the English top flight, however, having been drawn to face Norwich City after their upset win over Wolverhampton Wanderers.
The ties will be played on the weekend of January 26-27.