Racist slurs and pictures of the Egyptian forward by the man, 32, who claimed to be a Blues fan, led to him being slammed on social media.
A mocked-up image of the player was sent on Wednesday afternoon and was shared by other Twitter users.
The man, from Sefton, on Merseyside, has been held on suspicion of committing a racially aggravated public order offence and obstructing an officer in the execution of their duty.
Everton are investigating whether the message was sent by a Twitter user who had connections to the club.
“Merseyside Police is aware of an offensive tweet posted in relation to a Liverpool footballer, and shared by others, and will be investigating the matter”
A police spokesman said: “Merseyside Police is aware of an offensive tweet posted in relation to a Liverpool footballer, and shared by others, and will be investigating the matter.
“Hate crime in any guise will not be tolerated and those who use the internet to target others, and who commit a criminal offence such as hate crime in doing so, need to understand that they are not beyond the law.”
Everton have said they are making an internal enquiry to see if the man has any links to the club.
A spokesman said: “Everton condemns in the strongest terms any kind of racism.
“We have shared the material with the appropriate authorities and are looking into the matter further to understand if the Twitter user in question is known in any way.”
It comes as Everton fans have shown their support for new signing Moise Kean who suffered racist abuse in Italy.
Fan group County Road Bobblers have started a JustGiving campaign to raise money for a banner at Goodison Park to show him racism is not tolerated at the club.