Fifteen minutes into the Premier League match, racist language was heard in the Lower Gwladys Street Stands at Goodison Park.
In a statement, the club said: "Everton condemns all forms of discrimination and we will seek the strongest possible sanctions to tackle anti-social and or criminal behaviour, including club bans and criminal prosecution.
"If anyone has any information or footage that may assist our enquiries, this can be reported anonymously by texting HELP and the details of what was witnessed and where to 84300, or by emailing email@example.com."
The match itself ended with Everton losing 1-0 to Arsenal, with Leandro Trossard scoring the only goal in the 69th minute.
The club currently sits near the bottom of the Premier League in 18th place, with only one point from their first five games.