Morata, 28, revealed this week that he had received death threats and his family had received abuse when attending matches in Seville after the forward missed chances in the national team's three group games.
"It's so serious that it should be put in the hands of the police," Luis Enrique said in a news conference on Sunday ahead of Spain's round of 16 match with Croatia in Copenhagen. "Insulting and making death threats to anyone, let alone a family member or kids, is an offence, a serious offence... It should be put in the hands of the authorities and dealt with emphatically."
Morata was whistled by a section of the home crowd at Seville's La Cartuja stadium after failing to score during a 0-0 draw with Sweden in Spain's opening Euro 2020 game.
"I understand criticism for not scoring. I'm the first to accept it," Morata told Cadena COPE on Thursday. "I wish people could see what it is to receive threats and insults to your family, 'I hope your children die'... [Memes] don't bother me, what annoys me is what my wife has to go through. My kids go to Seville with their dad's name on their shirts."
Coach Luis Enrique has consistently defended the Juventus forward, pointing to his record for Spain -- with 20 goals in 43 international appearances -- and comparing it favourably with players such as Kylian Mbappe, Robert Lewandowski and Romelu Lukaku.
"We're exposed to all kinds of criticism and we accept it. But we don't accept any threats to a player, family members or kids," midfielder Koke -- a teammate of Morata with Spain and club Atletico Madrid -- said on Sunday. "We're all responsible for our actions and for what we say. Any type of harassment must be reported [to the authorities]."
He added: "I don't know what kind of people can make these comments on social media. They can do a lot of harm, not only to the player, you deal with it, but with family or kids that goes too far and should be reported."
Morata has spoken openly about the challenges of coping with pressure in football, admitting that "[I] didn't sleep for nine hours after the [Poland] game."
"How can I help him? By being as normal as possible," Koke said. "To help him focus, although when you get threats it isn't easy. I gave him a big hug after the game because I care about him a lot.
"He's a friend more than a teammate, we've played together since we were kids at Atletico... It isn't easy being in his situation. What he needed most in that moment was a hug from his wife and kids. When we're sat together we're normal, when we have to laugh and joke we do it. He's a funny guy, he's always laughing. We try to change the subject a bit and focus on what we have to be focused on."