Referee Mark Halsey has made a complaint to Greater Manchester Police about insulting tweets directed at him.
Halsey drew criticism from Reds manager Brendan Rodgers after sending off Jonjo Shelvey for a first-half tackle.
Halsey told BBC Sport: "I can confirm that my family have made a complaint to the police."
He added: "I will be speaking to the police on Wednesday after I have returned from refereeing Southampton v Sheffield Wednesday."
Rodgers was angry after Halsey also awarded Manchester United a late penalty, which Robin van Persie converted.
"We could control our players' performance but what we can't control is decisions by the referee," Rodgers said.
Some Liverpool fans later took to Twitter to criticise Halsey's decisions, making reference to him dying of cancer.
The abuse was condemned by the match officials' organisation, Professional Game Match Officials Limited.
PGMOL said: "[We] abhor any abuse of match officials... in stadia or outside of it."
The PGMOL said psychological support was available to referees. "Our main concern is towards Mark and his family, and like with all match officials, there is a backroom team who are there for them," it said.
"This includes a sports psychologist, who has worked extensively with the select group, including Mark, over the past few years.
"Mark has already benefited from the support of all of his colleagues and they will continue to back him this week."
Referees' union Prospect has also come out in support of the referee, calling the tweets "beneath contempt".
Former Premier League referee Dermot Gallagher said: "I know Mark as a friend and ex-colleague, and certainly, he will be shaken by it. Some things cross the boundaries, and this has.
"You expect stick as a referee, that is the nature of the game, but I think this has crossed the boundaries."
Halsey was diagnosed with throat cancer in 2009 but returned to refereeing in March 2010.
He will be in charge of next Saturday's Premier League match between Fulham and Manchester City.
Meanwhile, the Football Association is not expected to take any action over reports of offensive chanting by a minority of fans after the end of Sunday's game.
It noted that both clubs had been active in trying to stop offensive chants.