The announcement comes after a controversial article, written by columnist Kelvin MacKenzie, was published on Friday.
The article compared midfielder Barkley, whose grandfather was born in Nigeria, to a "gorilla" and said that men with similar "pay packets" to Barkley in Liverpool were "drug dealers".
The football club said in a statement: “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 operations.
“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.”
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.
My message to the people's club - listen to the people. We will continue to make our feelings heard until you do. pic.twitter.com/WdImReHybT— Joe Anderson (@mayor_anderson) April 14, 2017
"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."
News UK reported that Mr MacKenzie was suspended from the paper for the "wrong" and "unfunny" comments.
The former editor of the paper had been in charge 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".
His article on Barkley was written in response to video footage that emerged of the footballer being repeatedly punched by a man in a Liverpool bar.
The piece was widely criticised, with the The Mayor of Liverpool reporting it to the police and accusing it of being "racist and prehistoric".
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."