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Trevor Burke QC, for West Ham defender Zouma, said his client received abuse that was "the worst" the football players' union had seen.
He said police were looking to prosecute those responsible, but said the strength of the abuse gave rise to concerns about Zouma's safety, despite stringent security measures already being in place at his home.
Content of the messages was not read out at Thames Magistrates' Court in East London on Tuesday, where Zouma admitted kicking his cat in a Snapchat video filmed by his brother, Yoan, although a redacted copy was shown to the district judge, Susan Holdham.
She imposed an order under Section 11 of the Contempt of Court Act, banning publication of the Zoumas' addresses, in full or in part. She said: “I can see a number of messages - vile, racial abuse.
“There are threats to kill. They refer to him and his children. The bar (on media reporting) is a high one, but I think it has been reached.
“I quite understand the necessity for open justice and for the press to report freely but in these circumstances I am prepared for the address of both defendants to be withheld.”
The court order came after Zouma pled guilty to kicking and slapping a cat.
The West Ham defender was faced with three offences under the Animal Welfare Act after a video emerged in February which appeared to show him kicking and slapping his cat.
Zouma’s brother Yoan, who plays for Dagenham and Redbridge, has also been accused of two offences.
The maximum sentence for animal cruelty cases in the UK is five years after being raised from six months in 2021.
Zouma, 27, was forced to apologise after harrowing footage was filmed and shared by his younger brother who admitted one offence during the hearing.
Yoan, 24, admitted one count of aiding, abetting, counselling or procuring his older brother to commit an offence.
Defending the West Ham player, Mr Burke QC said his client had been fined £250,000 by his club and lost a "substantial" sponsorship deal with Adidas. He had also not been picked for the French national team, he added.
Mr Burke said: “A Google search will always reveal the guilty plea and the video.”
District Judge Susan Holdham adjourned proceedings until the sentencing hearing on 1 June.