Merseyside Police are investigating "unacceptable" online threats made towards Everton duo Jordan Pickford and Richarlison after the 2-2 draw in the derby against Liverpool on Saturday.
Goalkeeper Pickford made a challenge on Liverpool's Virgil van Dijk which forced the defender to leave the field after five minutes. As a result, the defender suffered an anterior cruciate ligament injury.
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Richarlison was sent off late on for a reckless tackle on Thiago and the forward apologised for his actions after the match.
Both players received threatening messages on social media and police warned those responsible for the tweets will be brought to justice.
"The language used in the tweets is totally unacceptable and we are taking the reports extremely seriously," a Merseyside Police spokesman said on Monday.
"Those who use the internet to target others, and who commit criminal offences such as making threats or committing a hate crime in doing so, are not beyond the law.
"Detectives are making inquiries to identify the individuals who posted the tweets and where we identify offences we will take action to bring those responsible to justice.
Liverpool announced that Van Dijk will undergo knee surgery and is facing a battle to play again this season and feature at next year's Euros.
The Netherlands international said he is determined to return "fitter and stronger" from the setback ahead of Liverpool's first Champions League match against Ajax on Wednesday.