The 24-year-old German goalkeeper was in tears at the final whistle in Kiev after the Reds’ 3-1 defeat to Real.
The Spanish side’s first goal came when Karim Benzema charged down an attempted throw-out by Karius, and the goalkeeper then allowed a long-range Gareth Bale shot to slip through his grasp for the third.
Merseyside Police released a statement saying its officers were aware of “threats” made on social media to Karius.
Karius tweeted on Sunday afternoon: “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...
“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...
“Thank you to our unbelievable fans who came to Kiev and held my back, even after the game. I don’t take that for granted and once again it showed me what a big family we are. Thank you and we will come back stronger.”
Karius had earlier told talkSPORT: “I lost my team the game. I’m sorry for everyone – from the team, from the whole club – that the mistakes cost dearly. If I could go back in time, I would. I feel sorry for my team. I know I let them down.
“It’s very hard right now but that’s the life of a goalkeeper. These goals cost us the title, basically.”