MacKenzie was suspended after comparing the midfielder, whose grandfather was born in Nigeria, to a "gorilla".
He also wrote that men with similar "pay packets" to Barkley in Liverpool were "drug dealers".
Everton in a statement said: "The newspaper has to know that any attack on this City, either against a much respected community or individual, is not acceptable."
Merseyside Police are currently investigating whether MacKenzie's comments constitute a "racial hate crime".
The Sun apologised "for the offence caused" and added that it was "unaware of Barkley's heritage".
In February, Liverpool banned Sun journalists from its grounds over the newspaper's coverage of the Hillsborough disaster.
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