The club announced on Saturday that The Sun newspaper, whose former editor Kelvin MacKenzie wrote the column, would be banned from covering their games and press events.
Everton fans then gave Barkley a rapturous reception as he came on to the pitch for the home match against Burnley at Goodison Park and again when he scored in the 3-1 win.
The Sun has suspended MacKenzie and apologised for the column which they said made "unfunny" and "wrong" comments about the people of Liverpool.
The paper also said they had not been aware of the family heritage of Barkley, whose grandfather was born in Nigeria. MacKenzie too said he was unaware of Barkley's family background and denied his article was racist.
MacKenzie's column has been reported to the police by the mayor of Liverpool Joe Anderson, who has also issued a complaint to the press watchdog.
Merseyside police confirmed they have received a complaint about the article and that they were carrying out enquiries.
The Sun is already banned by Everton's city rivals Liverpool due to their coverage of the Hillsborough disaster in which 96 people lost their lives in 1989.
MacKenzie was editor of the newspaper during its coverage, which was widely criticised for blaming fans for the disaster and accusing them of poor behaviour. The Sun later apologised for the coverage.
Saturday marked the 28th anniversary of the Hillsborough disaster and Everton held a minute's applause in memory of the victims prior to the kick off of their Premier League game with Burnley.
Barkley, who was attacked in a Liverpool bar last Sunday in an incident which prompted the scathing column, was given a warm welcome by Everton fans and after he scored their second goal, was booked by the referee for celebrating with them.
Read the full article on eurosport.co.uk: Everton rally round Barkley after banning The Sun