Liverpool's German keeper was blamed by many fans for failing to block goals, including one which one sports commentator described as a "schoolboy save," reports Xinhua news agency.
The Liverpool Echo newspaper reported that hundreds of explicit messages sent on social media are being looked into by Merseyside Police after the Champions League.
The newspaper said Karius and his family are the subjects of disturbing hate messages after mistakes he made during the match against Real Madrid were credited with LFC's defeat.
Some of the more extreme posts also threatened Karius' girlfriend and some even said: "I hope you get cancer".
Merseyside Police, headquartered in Liverpool, confirmed they are looking into the messages as a matter of urgency.
A spokesperson for the police said in a statement that "officers are aware of a number of comments and threats made towards football players via social media.
"The force takes social media posts of this nature extremely seriously and any offenses identified will be investigated," the statement continued.
"Merseyside Police would like to remind social media users that any offenses including malicious communications and threatening behaviour will be investigated.".
Karius used several social media messages on Sunday to apologise to his teammates and Liverpool fans.
In one he said: "Haven't really slept until now... the scenes are still running through my head again and again. I'm infinitely sorry to my teammates, for you fans, and for all the staff. I know that I messed it up with the two mistakes and let you all down."
In a second message, Karius said: "As I said I'd just like to turn back the time but that's not possible. It's even worse as we all felt that we could have beaten Real Madrid and we were in the game for a long time."
Meanwhile, an online petition, started in Britain, calling for Real Madrid player Sergio Ramos to be punished for "intentionally" injuring Liverpool star Mo Salah had received more than 150,000 signature by Sunday night.
Salah had to go to the hospital after a tackle 30 minutes into the game, prompting many Liverpool fans to say the injury seemed intentional.