Carragher said on Monday Night Football (h/t the Mirror's Mark Jones):
"He's the star player at Chelsea. But there has always been talk of him moving on.
"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.'
"Honestly, 100 per cent he would have been."
Chelsea lost 1-0 at the Etihad Stadium, but the scoreline did not accurately reflect City's dominance in the contest.
Per BBC Sport's Phil McNulty, the visitors had just 29 per cent possession and managed only three shots in the entire match, none of which were on target.
Former Chelsea player Mario Melchiot was not happy with the Blues' unambitious approach:
Mario Melchiot @MarioMelchiot
Today we played the same as a team that is trying to stay in the league. That’s not the right way! The last 15m was the bravest we have been. Why? Because we changed Hazard’s position from a striker to the right side and then take him off🤔 Killing his confidence. #mmlove #cfc
Carragher's fellow Monday Night Football pundit Gary Neville was also critical of what he saw:
On Hazard, Liverpool legend Carragher continued:
"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."
The winger himself expressed frustration at the match, though he did not say manager Antonio Conte was to blame, per football journalist Kristof Terreur:
Nevertheless, according to Matt Law of the Telegraph, Chelsea's players fear Carragher's assessment is correct and that regardless of Conte's future—Law noted he is expected to leave in the summer—Hazard will be eyeing a move to a club that can help him win the Ballon d'Or.
City and Real Madrid are said to be interested, and the Belgium international is yet to sign a new deal at Chelsea, who want to secure his future with a contract in excess of £300,000 per week.
Sunday's performance will have done little to convince him to stay.
The Blues have proved capable of significant improvement after disappointing campaigns—they won the title last year having finished 10th the season before—and should Conte leave, his replacement may be more positive in the biggest games.
However, at 27 and after six years at the club, it may not be surprising if Hazard is keen to explore a new challenge while he's still at his peak.