Carragher was unhappy at the Toffees appearing to mock former player Anthony Gordon.
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It happened after Saturday's 0-0 draw with Newcastle United at Goodison Park.
Gordon could have scored a penalty in the first half and had another great chance in the second half to win the game for his team.
In response, Everton's media team posted a video of Gordon's failed spotkick with mocking music on their official X account.
Carragher, response, posted on X: "Local lad who came through your academy...
“Helped keep you up under (Frank) Lampard, wanted out because the club was a shambles on & off the pitch & who you then sold for 40m. The Peoples club!"