Video report by ITV News Political Correspondent Carl Dinnen
Theresa May has told Donald Trump that sharing anti-Muslim videos posted online by far-right group Britain First "was the wrong thing to do".
But Mrs May indicated she is not withdrawing her invitation for the US President to come to the UK on a state visit, despite widespread calls for the trip to be cancelled.
Mr Trump sparked outrage in the UK by retweeting three videos from the far-right group, purportedly showing violent acts by Muslims.
In response to a statement from Downing Street describing his actions as "wrong", Mr Trump told Mrs May to "focus on terrorism" and not him.
But Number 10 hit back, saying Mrs May is "fully focused on dealing with extremism".
And speaking in Jordan on Thursday, Mrs May said: "The fact that we work together does not mean that we are afraid to say when we think that the United States have got it wrong and to be very clear with them.
"I am very clear that retweeting from Britain First was the wrong thing to do."
Asked whether she regarded Mr Trump as a fit person to be hosted by the Queen on a state visit, she said: "An invitation for a state visit has been extended and has been accepted. We have yet to set a date."
Mrs May described Britain First as "a hateful organisation" which "seeks to spread division and mistrust among our communities"
"As Prime Minister, I am very clear about the priority that I give to dealing with the challenge of the threat of terrorism...and extremism from whatever source they come," she said.
Britain's Ambassador to the US, Kim Darroch, said he complained directly with the White House over Mr Trump's tweets.
During a press briefing on Thursday afternoon, White House spokeswoman Sarah Huckabee Sanders said she did not believe Mr Trump knew who he was retweeting when he shared videos posted by Jayda Fransen, Britain First's deputy leader.
Asked if Mr Trump had elevated a far-right group to prominence, she added: "I think what he's done is elevate the conversation to talk about a real issue and a real threat and that's extreme violence and extreme terrorism, something that we know to be very real and something the president feels strongly about talking about and bringing up and making sure is an issue every single day."
Facing an Urgent Question in the House of Commons earlier on Thursday, Home Secretary Amber Rudd said Mr Trump was "wrong" to retweet the posts from Ms Fransen.
Video report by ITV News Correspondent Angus Walker
The question was brought by Labour MP Stephen Doughty, who said by sharing the tweets Mr Trump was "either a racist, incompetent or unthinking, or all three".
He called on the government to cancel Mr Trump's state visit, but the home secretary responded: "The invitation for the visit has been extended and accepted but the dates and precise arrangements have yet to be agreed."
After the prime minister's spokesman last night said Mr Trump's retweets were "wrong", the president lashed out, telling Theresa May: "Don’t focus on me, focus on the destructive Radical Islamic Terrorism that is taking place within the United Kingdom."
Asked about Mr Trump's Twitter rebuke to Mrs May, the Prime Minister's official spokesman told reporters: "Over her time as home secretary and as Prime Minister - and obviously in the wake of the tragic events over the summer - the Prime Minister is fully focused on dealing with extremism."
The US president sparked outrage on Wednesday after sharing three video posts by Ms Fransen, including footage from the Netherlands purporting to show a Muslim migrant committing crimes.
In the Commons, Mr Doughty told the home secretary: "This is the president of the United States, sharing with millions inflammatory and divisive content, deliberately posted to sow hatred and division, by, as the home secretary says, a convicted criminal who is facing further charges, who represents a vile, fascist organisation, who seeks to spread hatred and violence in person and online.
"By sharing it, he is either a racist, incompetent or unthinking, or all three."
Labour's shadow home secretary Diane Abbott said Mr Trump's actions were an "attack on the values of this country".
The prime minister's official spokesman said Britain First was dedicated to causing division among communities and that the president had been "wrong" to share the posts.
"It is wrong for the president to have done this," the spokesman said, adding that Britain First "cause anxiety to law-abiding people".
The spokesman said Britain First sought to divide communities through its use of "hateful narratives which peddle lies and stoke tensions".
"British people overwhelmingly reject the prejudiced rhetoric of the far-right which is the antithesis of the values that this country represents - decency, tolerance and respect," he added.
There has been some rare cross-party consensus following Trumps twitter behaviour.
Earlier, Mr Javid slammed Mr Trump for retweeting Britain First, which the Communities Secretary said "hates me and people like me".
Writing on Twitter, he said Mr Trump is "wrong" and he refuses to "let it go".
Foreign Secretary Boris Johnson also derided the group, describing Britain First as "divisive" and "hateful."
London Mayor Sadiq Khan, who has previously come under attack from the president on twitter, said in a statement: "Many Brits who love America and Americans will see this as a betrayal of the special relationship between our two countries.
"As the Mayor of this great diverse city, I have previously called on Theresa May to cancel her ill-judged offer of a state visit to President Trump," Mr Khan said.
"After this latest incident, it is increasingly clear that any official visit at all from President Trump to Britain would not be welcomed."
Ms Fransen, 31, from Penge, south-east London, is on bail facing four charges of causing religiously aggravated harassment as part of a Kent Police investigation into the distribution of leaflets and the posting of online videos during a trial held at Canterbury Crown Court in May.
She will also appear in court in Northern Ireland in December charged with using threatening and abusive language in connection with a speech she made at an anti-terrorism demonstration in Belfast on August 6.
Ms Fransen responded to the re-tweets by saying: "God bless you Trump! God bless America!"
Brendan Cox, widower of MP Jo Cox who was murdered by a right-wing extremist, said: "Trump has legitimised the far right in his own country, now he's trying to do it in ours. Spreading hatred has consequences & the President should be ashamed of himself."
Paul Joseph Watson, the UK-based editor of far-right conspiracy website Infowars, said: "Yeah, someone might want to tell whoever is running Trump's Twitter account this morning that retweeting Britain First is not great optics."
A spokesman for anti-fascism campaign group Hope not Hate said: "It beggars belief that a US president would share this material. Even Donald Trump.
"Anyone with half a brain knows that Britain First is an extreme right movement that has had links to Loyalism, claims to "invade" mosques, used murdered British soldier Lee Rigby's name against the wishes of his family to help it campaign in elections, and has bought many of its supposed followers on social media."
He added: "What on earth was Trump thinking, trying to propel it forwards?"
All three of the videos retweeted by Mr Trump from Ms Fransen are more than six months old, and two date back as early as 2013.
One video, shared by Britain First with the caption 'Muslim migrant beats up Dutch boy on crutches', is now the most popular from any of the group's twitter accounts.
It has received almost four times the retweets since Trump shared it.
Officials at the Dutch public prosecution issued a statement correcting the tweet and its caption.
The suspect in the video was born and raised in the Netherlands.
The suspect received a HALT settlement (restorative justice for juveniles).
Dutch police requested the video was removed at the request of the victim.
According to the De Telegraff newspaper, two 16-year-old boys were arrested and the boys' religions were not included in reports.
The other two videos, claiming to show violence and vandalism perpetrated by Muslims, date back to 2013 and 2015.
One shows an incident during riots in Alexandria, Egypt, in 2013, and another has very little contextual information to verify it.
Yesterday Sarah Huckabee Sanders told reporters she would not comment on the veracity of the videos.
In footage shared by broadcaster NBC News, she said: "I'm not talking about the nature of the video. I think you're focusing on the wrong thing."
She added: "The threat is real, the threat needs to be addressed, the threat has to be talked about and that's what the president is doing in bringing that up."
In July, a Press Association investigation found at least 10 misleading videos posted by the group to Twitter and Facebook over a two-month period.
Despite being alerted to the misleading nature of those videos, Twitter and Facebook declined to remove them.