The Croatian posted a direct message on Instagram he had received from a user that read: "I'm gonna murder ur family u Croatian prick."
The 28-year-old captioned it with "horrible what kind of people we have".
He added: "I don't mind when people talk shit about me, it says more about them! But I cannot ignore when my family is threatened. I just can't and won't accept that. Disgusting."
Liverpool manager Jurgen Klopp last week defended his player after substituting him just 31 minutes into a 4-1 loss at Tottenham Hotspur.
His performance was widely criticised but Klopp said: "I don't wish one of you to have your mistakes discussed in public. You cannot even imagine how it feels.
"The boys are in the first place still human beings but you look like you are watching an accident and you are the kind of people standing around with smartphones instead of helping. I'm not this kind of person.
"We had a normal week. Of course it's not the nicest week in Dejan's life but it's only football. People don't become a better or a worse person through making a mistake in a football game.
"If I think about Dejan, I have much more positive things than negative things."