The offensive article in question was published this week about Barkley following an incident in a nightclub in Liverpool in which columnist Kelvin MacKenzie called the midfielder "thick" and insinuated that the only people who could a wage similar to footballers in Liverpool were drug dealers.
The article also compared Barkley, who is mixed race, to a gorilla. News UK later announced MacKenzie had been suspended.
Anderson has responded to the piece by telling the Liverpool Echo that he feels "embarrassed” to be an Everton fan with the newspaper still not banned by the club.
Liverpool have banned The Sun from press conference and covering games at Anfield in response to the paper's coverage of the Hillsborough disaster of 1989.
Anderson also took to Twitter to call upon Everton to join Liverpool in this move.
He also added that he has reported The Sun and Mackenzie to the Merseyside Police and the Press Complaints Commission.
In his full statement to the Echo he said:
I think Everton have let the fans down and let the city down by not banning them. This weekend of all weekends, when we are trying to do something to remember the 96 fans who died at Hillsborough, at the same time Everton are allowing The S*n - which is still violently attacking our city - into its press conferences. I have asked (Everton chief executive) Robert Elstone behind the scenes before to act on this and I have been reluctant to do this publicly but now I have been forced to go public. For god’s sake, stop embarrassing the fans, they are attacking the city and the club is saying it’s ok - I feel embarrassed to be an Everton fan.
Anderson also tweeted his disappointment with the Sun after reports of Mackenzie's suspension surfaced.
Read the full article on eurosport.co.uk: The Sun reported to police over Ross Barkley article, MacKenzie suspended