Karius, 24, and his family have been the subject of threats after he made two errors in the final against Real, effectively costing Liverpool the game.
The Spanish side’s first goal came when Karim Benzema charged down an attempted throw by Karius, and the goalkeeper then allowed a tame long-range Gareth Bale shot to slip through his grasp for the third.
“We take social media posts of this nature extremely seriously. Offences will be investigated,” the police said in a short statement.
“Officers are aware of a number of comments and threats made via social media.
“Merseyside Police would like to remind social media users than any offences including malicious communications and threatening behaviour will be investigated.”