The Premier League champions drew a great deal of criticism for a tame performance after they went down 1-0 to Pep Guardiola's side on Sunday, their fourth loss in five top flight matches.
Sitting fifth in the table and 25 points behind City, Chelsea's struggles have only strengthened media speculation surrounding Hazard's future, with Real Madrid reportedly determined to lure him to the Spanish capital.
And Carragher feels the Blues are only pushing him away with performances like that at the Etihad Stadium, saying the Belgian will already be urging his agent to find him a new club.
"I have no doubt that at the end of that game, as soon as he got on that coach or got home, he will have been right on the phone to his agent saying, 'get me out of here'.
"He will have been that frustrated thinking, 'I should be playing in this team, they've got the ball. I should be playing in a team that actually dominates the game'.
"Eden Hazard running around up there on his own, not getting a kick of the ball. You want to show everyone in the world watching how good of a player he is.
"I think they will have a major problem keeping him after that."
Hazard, 27, is Chelsea's top scorer in the league this season with 11 strikes and three assists in 25 appearances. The ex-Lille star is contracted to the Stamford Bridge side until 2020.