Coleman was in the spotlight after he conceded a penalty in Saturday's 2-1 defeat to Reading and that sparked a nasty reaction from some so-called Everton supporters, with a number of anti-Irish comments.
The incident comes just days after police in England announced that they were investigating online death threats made against Coleman's Ireland team-mate James McClean over his refusal to wear a poppy.
Everton are taking the matter seriously. "We 100 per cent abhor every single one of those comments made by mindless individuals who have shown no respect for the strong links Everton has with its Irish supporters," an Everton spokesman told the Donegal News.
"As a football club we have enjoyed a strong relationship with Irish players throughout the years and we enjoy a great support throughout the country.
"Unfortunately, we live in an age where social media now allows these faceless individuals the opportunity to abuse people like this as and when they feel like it."