Manuel Pellegrini's men are 1-10 to ease past Rovers at the second attempt and face a fourth-round clash with Bristol City or Watford.
Arsenal (9-2), Chelsea (5-1) and Liverpool (5-1) all find themselves next in the betting, while Everton are looking to get their hands on the famous trophy for the first time in nearly 20 years at odds of 9-1.
West Ham's thrashing from Nottingham Forest has done nothing but ease the thoughts of Sam Allardyce's backers to be sacked next and the result has seen his odds shorten even further to just 4-11 at Ladbrokes.
Jessica Bridge of Ladbrokes said: “Man United's exit won't shock many punters these days as they're no longer a reliable coupon banker. City remain slight favourites despite still needing to see off Blackburn.
“Big Sam's days look numbered and it looks like we'll be paying out on his exit sooner rather than later.”