It comes with relations between British Prime Minister Theresa May and Trump deteriorating in a public spat over Trump's tweeting of videos posted by Britain First, a far-right group.
May used her first public comments on the matter to rebuke Trump, saying he was "wrong" to share the videos and insisting her cabinet ministers would never do the same.
May denounced Britain First as a "hateful organisation" that "seeks to spread mistrust and division within our communities".
She also appeared to question the US record of keeping the far-right in check and poked fun at Trump's love of Twitter by saying she rarely looked at the social media platform.
Sir Kim Darroch, the British ambassador in Washington, has formally complained to the White House about Trump's behaviour.
The row is the most serious breakdown in trust between May and Trump since he took office this year and throws into question her decision to get close to the US President.
It began when Trump shared three anti-Muslim videos posted by Jayda Fransen, the deputy leader of Britain First, with his 43 million followers on Twitter.
They purported to show a "Muslim migrant" attacking a Dutch boy, a Muslim destroying a Virgin Mary statue and an "Islamist mob" pushing a teenager off a roof.
Some of the information in the messages proved incorrect and the tweets drew condemnation from Cabinet ministers and May's own spokesman.
Trump hit back, tweeting later on Thursday: "@Theresa-May, don't focus on me, focus on the destructive Radical Islamic Terrorism that is taking place within the United Kingdom. We are doing just fine!"
The comment escalated calls for May to rescind the offer of a formal state visit to Trump, which was made in January but is yet to take place.
Asked whether Trump had an obligation to verify information before posting it on Twitter, his press secretary Sarah Huckabee Sanders said: "I think what he's done is elevate the conversation to talk about a real issue and threat.
"And that's extreme violence and extreme terrorism."
May doubled down on her Government's criticism of Trump during a visit to Jordan yesterday, saying: "I am very clear that retweeting from Britain First was the wrong thing to do."
Asked about the group and whether Trump was legitimising them, May said: "I think that we must all take seriously the threat that far-right groups pose, both in terms of the terrorist threat that is posed by those groups and the necessity of dealing with extremist material which is far-right as well.
"I have commented in the past on issues in the United States on this matter. In the United Kingdom we take the far-right very seriously and that is why we ensure that we deal with these threats and this extremism wherever it comes and whatever its source."
May said that Britain would not be afraid to rebuke the US, despite the so-called special relationship between the countries. "The fact that we work together does not mean that we are afraid to say when we think," she said.
Meanwhile, Sadiq Khan, the London Mayor, said Trump would not be welcome in the capital.
He said: "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.
"After this latest incident, it is increasingly clear that any official visit at all from President Trump to Britain would not be welcomed."
In the US, senior Republicans such as Orrin Hatch, the senator for Utah, and Lindsey Graham, the senator for South Carolina, backed May and criticised Trump.
