Joe Anderson took offence at remarks made by the newspaper's former editor about the footballer who was punched in a Liverpool city centre bar in the early hours of Monday.
The columnist has also been suspended from the newspaper after he expressed "wrong" and "unfunny" views about the people of Liverpool, News UK reported.
In his column on Friday, Mr MacKenzie wrote: "Perhaps unfairly, I have always judged Ross Barkley as one of our dimmest footballers.
"There is something about the lack of reflection in his eyes which makes me certain not only are the lights not on, there is definitely nobody at home.
"I get a similar feeling when seeing a gorilla at the zoo. The physique is magnificent but it's the eyes that tell the story.
"So it came as no surprise to me that the Everton star copped a nasty right-hander in a nightclub for allegedly eyeing up an attractive young lady who, as they say, was 'spoken for'.
"The reality is that at £60,000 a week and being both thick and single, he is an attractive catch in the Liverpool area, where the only men with similar pay packets are drug dealers and therefore not at nightclubs, as they are often guests of Her Majesty."
The 23-year-old England midfielder has a Nigerian grandfather.
Labour councillor Mr Anderson told BBC Sport: "Not only is it racist in a sense that he is of mixed-race descent, equally it's a racial stereotype of Liverpool. It is racist and prehistoric."
He later tweeted: "Reported McKenzie&the S*n for their racist slur on Ross Barkley and the people of Liverpool to Merseyside Police&press complaints commission."
Mr MacKenzie oversaw the newspaper when it ran a front page story four days after the 1989 Hillsborough tragedy, headlined The Truth, which featured claims that some Liverpool fans had "picked pockets of victims" and "urinated on brave cops".
He has issued several apologies since over the coverage which he now says was based on "deliberate misinformation from the South Yorkshire Police".
Last year an inquest jury concluded the 96 fans were unlawfully killed and that the behaviour of Liverpool supporters at the FA Cup semi-final played no role in causing the tragedy.
Mr MacKenzie told the Press Association: "I had no idea of Ross Barkley's family background and nor did anybody else.
"For the mayor of Liverpool and a handful of others to describe the article as racist is beyond parody."
In a statement, News UK said: "The Sun apologises for the offence caused.
"The paper was unaware of Ross Barkley's heritage and there was never any slur intended.
"Mr MacKenzie is currently on holiday and the matter will be fully investigated on his return."