In an interview with Italian daily la Repubblica, organised by the Foreign Press Association, Dame Cressida said: “The racist abuse that we saw of the footballers after last Sunday was vile and horrible.
“It is being investigated. Our National Football Crime Unit is coordinating those investigations.
“You may already have seen that some people in the UK have been arrested and I’m sure that will continue.”
Marcus Rashford, Bukayo Saka and Jadon Sancho’s social media accounts were flooded with racist messages after the trio missed penalties in the final of Euro 2020 at Wembley.
The FA later released a statement condemning the “disgusting behaviour”. Prime Minister Boris Johnson also attacked the “appalling abuse”.
It comes as police hunt 10 men over incidents at Wembley on the night of the final.
Dame Cressida emphasised that cases to bring to court online criminals from overseas could be more complicated.
“That’s certainly a challenge. Sometimes they’re not and we have many examples of people being brought to justice for online hate crime and racist abuse in this country,” she explained.
She also piled pressure on social media companies to do more to tackle the scourge of racism.
“There is a whole series of conversations by governments with the online platforms about what more they can do to make the spaces safe for people, what more they can do to remove offensive content as soon as it is there, and what more they may be able to do to assist law enforcement, to investigate and to provide material for law enforcement from the platform itself or devices that we may see,” she said.
She highlighted the Online Safety Bill which could force social media firms to take tougher action.
“This is probably, I believe, maybe one of the first in the world. It’s quite comprehensive,” she said.
“ I’m sure it will be strongly debated because this is about the balance between people’s privacy, people’s freedoms of speech and things which are frankly, you know, appalling, hateful, shouldn’t be allowed, and criminal.”
People found guilty of online racist abuse are to be banned from attending football matches under Government plans.