The abuse aimed at Alvaro Morata and his family by some Spain supporters crossed the line and is "not acceptable", according to team-mate Dani Olmo.
Morata has been the focus of attention for La Roja since being jeered by his own fans during a pre-Euro 2020 friendly with Portugal after missing several opportunities.
He was then criticised for his displays in Spain's 0-0 draw with Sweden in their Group E opener and 1-1 draw with Poland, a game in which he scored but missed more chances.
The Juventus striker, whose loan from Atletico Madrid was extended for another season last week, then missed a penalty in Wednesday's 5-0 win against Slovakia.
Morata revealed in an interview that he has received vicious messages on social media during the tournament, while his wife and children have also been targeted in public.
"I would like people to put themselves in my shoes and think what it's like to get threats towards my family, people saying: 'I hope your children die'," he told Cadena Cope.
"I've had to leave my phone outside my room. My wife and children have come to the stadium in Seville with Morata on the back of their shirts and people have been shouting at them.
"It's complicated. I understand people booing me for missing chances but there's a limit."
Morata has had five big chances at Euro 2020 so far. Only Cristiano Ronaldo (six) has had more, though the Portugal skipper has scored five times.
The former Chelsea striker's four Opta-defined big chances missed is the most of any player in the tournament, followed by Gerard Moreno and seven others on three misses.
Spain boss Luis Enrique has repeatedly leapt to the defence of Morata for his performances and is expected to stick with the 28-year-old for the last-16 tie with Croatia.
And team-mate Olmo, who started two of the group games, feels the vitriol Morata has faced has gone too far.
"Myself and all the team are against the threats," he said at a news conference on Saturday previewing Spain's knockout clash with Croatia on Monday.
"It is normal that we are criticised, it is part of our sport, football, and we are exposed to it. We accept it. But going beyond it is not acceptable.
"I think people should put themselves in the skin of the player. We all are human being and we need to respect it.
"Again, I understand if we are criticised as it is part of our profession, but going beyond it is too much."
After winning Euro 2008, World Cup 2010 and Euro 2012, Spain have since been eliminated from both of their last two major tournament knockout-round matches.
Croatia won the most recent of their meetings with Spain 3-2 in November 2018, a year before Olmo earned his first senior cap for La Roja.
Olmo spent his formative years in Dinamo Zagreb's academy and could have represented Croatia, but he instead pledged allegiance to Spain.
"It's true there was an interest from Croatia, but I believe I have always made clear my side," he said.
"I was thankful for that interest as I have special love for Croatia, but I always said my dream was to be here with Spain. At the end, I got it, and I am where I always wanted to be.
"Croatian people have showed to be very competitive in all sports, not only in football.
"It is a country not as big as Spain, France or Germany, but they compete even more in every single sport. And they are very patriotic. They give their all for their country."
Asked if head coach Luis Enrique has asked for any inside information on Croatia, Olmo said: "Not yet! I think he knows their team very well.
"I also obviously know them very well and have played with many of them. If he asks me something, I will give him my opinion, as usual.
"I've spoken to a couple of their players already and cannot wait to play the game."
Hurtig the hero after late goal sends Arsenal into Champions League quarters
Lina Hurtig's 89th minute winner ensured Arsenal progressed to the quarter-finals of the Women's Champions League after edging Juventus 1-0 on Thursday.
After being frustrated for large parts of the encounter, Hurtig bundled home from close range after Martina Lenzini's poor clearance to book the Gunners' place in the last eight.
Despite Arsenal dominating the early exchanges, it was Juventus who had the first sight of goal, though Eva Schatzer's effort was easy for Daphne van Domselaar to gather.
In a first half that saw just two shots on target, both coming from the visitors, Renee Slegers' substitutes proved to be the difference for Arsenal.
Slegers introduced Beth Mead, Stina Blackstenius and Hurtig late on, with the latter somehow putting Kim Little's cross over the bar from five yards out.
However, the Swede would atone for her earlier error, combining with fellow substitute Blackstenius to net against her former club and send them out of the competition with two group games remaining.
Meanwhile, in Group D, Barcelona edged closer to the quarter-finals of the competition with a convincing 4-1 victory over St. Polten at the Generali Arena.
The reigning champions are still three points behind Manchester City following their 2-1 win over Hammarby, and know a win against the Swedes next month will see them progress.
Kika Nazareth opened the scoring in the 20th minute after capitalising on a mistake by Kamila Dubcova, notching her second nine minutes later from close range.
Barca found themselves three goals up at the break when Vicky Lopez netted her first goal in the competition to put the result beyond doubt.
There was also a landmark moment for Alexia Putellas, who netted her 200th goal for the club with a well-taken effort into the far corner.
Valentina Madl took some of the shine off an otherwise impressive display from the visitors, who have now taken their tally to 20 goals from their four matches in the competition.
Guardiola agrees two-year contract extension with Man City
Pep Guardiola will be Manchester City manager for at least two more seasons after 2024-25, after he signed a two-year contract extension on Thursday.
The 53-year-old was coming to the end of his City deal, which was due to expire at the end of the current campaign, but he has opted to remain at the Etihad Stadium.
The deal itself will take Guardiola's tenure at the club to over a decade, making him City's longest-serving manager since Les McDowall in 1963.
There had been suggestions that he may leave at the end of the season following the announcement that director of football Txiki Begiristain was on his way out.
Guardiola was strongly linked with the England job after Gareth Southgate resigned in the wake of their Euro 2024 final defeat, but the Football Association appointed Thomas Tuchel last month.
“Manchester City means so much to me," Guardiola said. “This is my ninth season here; we have experienced so many amazing times together. I have a really special feeling for this football club.
“That is why I am so happy to be staying for another two more seasons.
“Thank you to everyone for continuing to trust and support me – The Owner, The Chairman Khaldoon, Ferran, Txiki, the players and, of course, the fans… everyone connected to Manchester City. It has always been an honour, a pleasure and a privilege to be here.
“I have said this many times before, but I have everything a manager could ever wish for, and I appreciate that so much.
“Hopefully, now we can add more trophies to the ones we have already won. That will be my focus.”
Having been appointed in 2016, Guardiola is by some distance Manchester City's most successful manager of all time, having won 18 major trophies during his time at the club.
That list includes their first-ever Champions League triumph in 2023 and six Premier League titles, including an unprecedented four on the spin.
He also boasts a 72.04% win percentage across all competitions, which is unsurprisingly the highest of any full-time City manager in history.
Shaw at the double as Man City progress in Champions League
Khadija Shaw scored twice as Manchester City beat Hammarby 2-1 to seal their place in the Women's Champions League quarter-finals.
Gareth Taylor's team entered Thursday's game with a perfect record in Group D, and their fourth win in as many European matches ensured they finished in the top two.
Shaw—the leading scorer in the Women's Super League this season with eight goals—put them ahead in the 31st minute, driving into the area after taking in Yui Hasegawa's pass before seeing her shot deflect into the top corner.
Shaw was booked for celebrating in front of the home fans despite appearing to have objects thrown at her from the stands, and Hammarby should have levelled when Julia Blakstad fired wide 10 minutes later.
The hosts did draw level three minutes into the second half as Ellen Wangerheim tapped home while City's Laura Blindkilde Brown was down appealing for a free-kick.
That parity was short-lived, though, as Shaw blasted into the top corner from 25 yards out, taking her overall tally for this season to 12 goals in all competitions.
In Group C, meanwhile, Bayern Munich lost their perfect record as they played out a 1-1 draw with Valerenga, boosting Arsenal's hopes of topping the pile.
Jovana Damnjanovic came off the bench to put Bayern ahead 15 minutes from time in Norway, but Elise Thorsnes headed home following an 88th-minute corner to snatch a point for the hosts.
Arsenal could now move within a point of Bayern when they face Juventus in Group C's other match later on Thursday.
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