The alleged "racial slur" against Barkley centred on MacKenzie's opinion that he was a "one of our dimmest footballers". The columnist added: "I get a similar feeling when seeing a gorilla at the zoo."
But the columnist said he had "no idea" of the 23-year-old England international's heritage, as Barkley has a Nigerian grandfather.
Mayor Anderson reported the comments to Merseyside Police and IPSO - the Independent Press Standards Organisation - while MacKenzie was suspended by News UK.
The mayor, 59, told Radio 4's Today programme: "He's despicable. He deliberately smeared our city so let's hope that The Sun editorial board or the owners of News International [now News UK] do the right thing now and sack him, because it's exactly what he deserves."
The column was in response to a video showing Barkley being punched in a bar in the city following their 4-2 win over Premier League champions Leicester City last Sunday.
However, it comes at a time when emotions are running high in Liverpool as today is the 28th anniversary of the Hillsborough Disaster, where 96 Liverpool fans were crushed to death in English football's worst ever tragedy.
The Sun newspaper, which MacKenzie edited at the time, ran a front page article in the aftermath headlined "The Truth", claiming the club's supporters "picked pockets of victims" and "urinated on police". The allegations, which the paper and MacKenzie have since apologised for, has seen newsagents in the city refuse to stock the paper ever since.
Following his suspension, MacKenzie, 70, said: "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."
A News UK spokesman, confirming MacKenzie's suspension, said the company found the views of people from Liverpool expressed by MacKenzie "wrong" and "unfunny".
He added: "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."
In a statement outlining the club's reasons for banning The Sun newspaper, Everton FC said: "Yesterday Everton Football Club informed The Sun newspaper it was banned from Goodison Park, the USM Finch Farm training ground and all areas of the Club's operation.
"Whilst we will not dignify any journalist with a response to appalling and indefensible allegations, the newspaper has to know that any attack on this City, either against a much respected community or individual, is not acceptable."