Boris Johnson has warned violent protesters they will "face the full force of the law" after claiming Black Lives Matter demonstrations are likely to "end in deliberate and calculated violence".
He said it is "absurd and shameful" the statue of Sir Winston Churchill in Parliament Square has to be boarded up ahead of protests this weekend.
The Cenotaph and the Churchill statue in Westminster were defaced and spray painted during a Black Lives Matter march through central London.
"A growing minority unfortunately have hijacked them and they are using them as a pretext to attack the police to cause violence and to cause damage to public property.
"My message to everybody is for all sorts of reasons, they should not go," the prime minister said.
"You should not have a situation where people who are protesting on one basis are violently attacking the police and public property."
He acknowledged the outrage people feel about racism but said the UK has made huge progress in fighting it.
“I think this is a country that has made huge progress in tackling racism.
“We should look sometimes at the positive stuff – we’ve got more young black and minority ethnic kids going to university than ever before, more black kids doing the tougher subjects at school, doing better than ever before in school," he said.
Protective fences have been placed around the memorials, along with statues of Gandhi and Nelson Mandela, as the capital prepares for protests from the far-right and anti-racism demonstrators.
Mr Johnson said police have “already made hundreds of arrests in the last few days and they will make many more".
He said police "can see the culprits" because "overwhelmingly they are being recorded with body worn cameras".
The Metropolitan Police have said they are ready to tackle violence directed at police or property following disorder at protests last weekend.
Commander Bas Javid said both Black Lives Matter protesters and right-wing groups will be dealt with using the same tactics but urged people to make their voices heard in other ways, such as social media.
He said: “Do not come to London to protest in large groups because not only is that unlawful, but you’re putting yourself and other people at risk from a health perspective."
Speaking to broadcasters, Mr Johnson said: "We should not support a demonstration that is in all probability, looking at what has happened before, going to end in deliberate and calculated violence."
Writing on Twitter, the prime minister said: "The statue of Winston Churchill in Parliament Square is a permanent reminder of his achievement in saving this country - and the whole of Europe - from a fascist and racist tyranny.
"It is absurd and shameful that this national monument should today be at risk of attack by violent protesters. Yes, he sometimes expressed opinions that were and are unacceptable to us today, but he was a hero, and he fully deserves his memorial.
"We cannot now try to edit or censor our past. We cannot pretend to have a different history. The statues in our cities and towns were put up by previous generations.
"They had different perspectives, different understandings of right and wrong. But those statues teach us about our past, with all its faults. To tear them down would be to lie about our history and impoverish the education of generations to come."
Mr Johnson added the "only responsible action" was to stay away from planned protests this weekend.
"As for the planned demonstrations, we all understand the legitimate feelings of outrage at what happened in Minnesota and the legitimate desire to protest against discrimination.
"Whatever progress this country has made in fighting racism - and it has been huge - we all recognise that there is much more work to do.
"But it is clear that the protests have been sadly hijacked by extremists intent on violence. The attacks on the police and indiscriminate acts of violence which we have witnessed over the last week are intolerable and they are abhorrent.
"The only responsible course of action is to stay away from these protests."
Officials anticipate the memorials could be a hot spot of clashes between anti-racism protesters and counter-demonstrators who have vowed to protect the sites.
The Democratic Football Lads Alliance called on supporters to travel to London to protect monuments after a number were vandalised in recent protests.
Far-right figure Tommy Robinson, whose real name is Stephen Yaxley-Lennon, has expressed his support.
A Black Lives Matter protest due to take place in Hyde Park at 1pm on Saturday has reportedly been called off after organisers warned that "many hate groups" were targeting those planning to attend.
London mayor Sadiq Khan said he was "extremely concerned" about further protests in London, particularly by extreme far-right groups which “advocate hatred and division”, could lead to violence and disorder.
Mr Khan said: “It is clear that the majority of the protesters have been peaceful. This moment must be a catalyst for systemic, lasting change to tackle the racism and inequalities that black people still face today, in this country and elsewhere.
“However, I’m extremely concerned that further protests in central London not only risk spreading Covid-19, but could lead to disorder, vandalism and violence.
He added: “Staying home and ignoring them is the best response this weekend.”
Mr Khan said he is working with the Metropolitan Police and partners to ensure statues and monuments at risk, including the Cenotaph, Winston Churchill and Nelson Mandela, are covered and protected.
Scaffolding and a wooden covering was placed around the statue of Winston Churchill on Thursday evening, with the memorial in Parliament Square fenced off.
Black Lives Matter protests have sparked debate in the UK about the position of historical figures associated with slavery and racism.
In Bristol, protesters tore down the statue of slave trader Edward Colston while dozens of other petitions and movements have demanded the removal of other statues linked to slavery across the country.
Oxford University's statue of Cecil Rhodes has once again become a focal point of anger among protesters for his links to the slave trade.
In east London, the statue of slave owner Robert Milligan was taken down. And in Plymouth, a public square named after a 16th century slave trade is to be renamed, the city council said.
While in Poole, supporters of a controversial statue camped out overnight on Thursday to ensure the memorial was not torn down and thrown into the sea.
Robert Baden-Powell had expressed sympathy towards Hitler and the Nazis when he was alive but supporters of the statue say it must stay put.