The 24-year-old German and his family have been the focus of social media threats and hate messages after Saturday's match in Kiev, Ukraine.
"The force takes social media posts of this nature extremely seriously and any offenses identified will be investigated," said a statement from Merseyside Police, which covers the Liverpool region.
"Merseyside Police would like to remind social media users than any offenses including malicious communications and threatening behavior will be investigated."
Karius 'infinitely sorry'
The German had an evening to forget in the Ukrainian capital, with Real scoring from his errors in the 51st and 83rd minute.
Indeed, the final in the Olimpiyskiy Stadium will be remembered as much for Karius' dreadful mistakes as the spectacular bicycle kick from Gareth Bale which put the Spaniards 2-1 ahead.
Karius was distraught at the final whistle and posted on social media Sunday that he was "infinitely sorry" to his teammates, Liverpool fans and staff.
"Haven't really slept until now. The scenes are still running through my head again and again," the goalkeeper wrote.
"I know I messed it up with the two mistakes and let you all down.
"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."
With the match goalless early into the second half, Karius picked up the ball on the edge of his penalty area and appeared to try to roll the ball to a teammate, seemingly unaware of the lurking Karim Benzema, who intercepted to score into an empty net.
At 2-1, the goalkeeper fumbled Bale's long-range shot to leave the result in no doubt.
As Real celebrated at the final whistle, an inconsolable Karius buried his face into the turf and then tearfully apologize to Liverpool's supporters.
Teammates have since come out in support of their goalkeeper.
"When we win, we win together. When we lose, we lose together. Don't blame him," Liverpool defender Dejan Loren told reporters.
"It's easy to blame someone but we are in the same ship together and everyone gave him the best words that they can. He will come back strong. You cannot find the words."
Salah 'confident' of making World Cup
was another Liverpool player in tears Saturday after injury ended the Egyptian international's participation before halftime.
Salah -- who has scored 44 times in a record-breaking debut season
for the Reds - sustained a shoulder injury during a collision with Real captain Sergio Ramos.
On Sunday the forward eased fears that he could miss the forthcoming World Cup in Russia.
"It was a very tough night, but I'm a fighter. Despite the odds, I'm confident that I'll be in Russia to make you all proud," he posted on Twitter.
"Your love and support will give me the strength I need."