Messages referencing the 51-year-old's battle with throat cancer were posted on the social networking website after he officiated Manchester United's win over Liverpool
Several tweets regarding the veteran official's battle with throat cancer in 2009 were posted after a game in which the 51-year-old sent off Reds midfielder Jonjo Shelvey and handed the visitors a penalty.
"I can confirm that my family have made a complaint to the police," Halsey told BBC Sport.
"I will be speaking to the police on Wednesday after I have returned from refereeing Southampton v Sheffield Wednesday."
Professional Game Match Officials, who provide referees for Premier League matches, have also condemned the messages and offered their support to Halsey and his family.
"PGMO abhors any abuse of match officials whether that is in stadia or outside of it," read a statement released to Press Association Sport.
"Our main concern is towards Mark and his family, and as with all match officials, there is a backroom team who are there for them.
"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 Select Group colleagues and they will continue to back him this week."
PGMO went on to confirm that the messages were not sent directly to Halsey, stating: "None of the Select Group match officials are on social media.''