Merseyside Police are investigating racist comments posted on social media following the incident between Liverpool's Roberto Firmino and Everton's Mason Holgate in Friday night's FA Cup tie.
The pair were involved in a heated exchange when Firmino reacted to being pushed into the stands by Holgate, and the Everton defender's Twitter account has since been deleted amid an FA probe into the incident.
"We can confirm we are aware of racist comments posted by a number of individuals on Twitter on January 5, 2018, and will be investigating this,'' Merseyside Police said on Twitter on Sunday evening.
Deputy chief constable Carl Foulkes added: "We are working with Liverpool FC and Everton FC and hate crime will not be tolerated.''
The Football Association launched its own enquiry on Saturday after Bobby Madley included the incident and subsequent allegation, made by Holgate on the pitch and Everton officials afterwards, in his match report.
"The FA can confirm that referee Bobby Madley was made aware of an allegation during the Liverpool versus Everton game at Anfield [on Friday night] and has subsequently reported this to the FA, which will now begin making enquiries into the matter,'' a statement from the FA read.
Immediately after the match, Liverpool pledged to fully co-operate in any investigation.
Press Association Sport reports Firmino was not questioned about the incident after the game but is prepared to provide a statement should he be requested to by the FA.
Both Everton manager Sam Allardyce and Liverpool counterpart Jurgen Klopp refused to comment on the incident after the match.
Liverpool went on to win the third-round tie 2-1 at Anfield.