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| owning Street has said David Cameron stands by his criticism of Donald Trump but will work with whoever is elected US president. |
he UK PM has called the Republican hopeful "stupid, divisive and wrong" over his call for a temporary ban on Muslims entering the US.
Mr Trump replied by warning he may not have a "very good relationship" with Mr Cameron if he enters the White House.
He is also involved in a spat with new London Mayor Sadiq Khan.
The US presidential contender said he would not forgive Mr Khan for calling him "ignorant" - and challenged the Mayor to take part in an IQ test, an offer mocked by Mr Khan's team.
Last year, Mr Trump, who has beaten his rivals to become the presumptive Republican candidate, called for a temporary halt to all Muslims entering the US in the wake of the deadly terror attack in San Bernardino, California.
He said many Muslims nursed a "hatred" towards America and a ban should be in force "until our country's representatives can figure out what is going on".
Responding to Mr Trump's comments at the time, Mr Cameron said: "I think his remarks are divisive, stupid and wrong and I think if he came to visit our country I think it'd unite us all against him."
Speaking on ITV's Good Morning Britain earlier on Monday, Mr Trump said: "It looks like we're not going to have a very good relationship, who knows.
"I hope to have a good relationship with him, but it sounds like he's not willing to address the problem either."
Mr Trump said he was not anti-Muslim but "anti-terror", saying "we have a real problem and we have to discuss it", as he called on Muslims to work with the people and "turn people in" that they suspected of extremism.
"I have many Muslim friends," he said. "I was with one the other day, one of the most successful men, he's Muslim and he said, 'Donald you have done us such a favour, you have brought out a problem that nobody wants to talk about'."
'Special relationship'
David Cameron's official spokesman was asked about Mr Trump's suggestion that they might not have a good relationship following the Republican candidate's latest comments.
"The prime minister has made his views on Donald Trump's comments very clear. He disagrees with them and I haven't got anything further to add," said the spokesman.
"He continues to believe that preventing Muslims from entering the US is divisive, stupid and wrong."
The Number 10 spokesman said that Mr Cameron was "committed to maintaining the special relationship" whoever wins the presidential election.
"He has been clear that he will work with whoever is president of the United States," said the spokesman.
No proposal had been made for a phone call between the prime minister and Mr Trump, but the spokesman said Downing Street would be willing to consider it.
IQ test
Mr Trump also criticised what he called the "very rude statements" made about him by Sadiq Khan, after Mr Trump suggested he would make an "exception" to the ban for the London mayor.
Mr Khan, the first directly-elected Muslim mayor of a major Western capital city, dismissed Mr Trump's offer and accused the US presidential hopeful of holding "ignorant" views of Islam which "could make both our countries less safe" by playing into the hands of extremists.
Responding, Mr Trump told ITV: "I am offended, he doesn't know me.
"I think they were very rude statements and, frankly, tell him I will remember those statements," he added.
Mr Trump also challenged Mr Khan to an IQ test.
'Back of the queue'
But as the public spat between the two men continued, a spokesman for Mr Khan said US voters would reject Mr Trump's "ignorant, divisive and dangerous" views.
He said there were "no plans" to seek direct talks with Mr Trump and mocked his IQ challenge, saying: "Ignorance is not the same thing as lack of intelligence."
Mr Khan, who was elected Labour mayor of London last week, told BBC News his message for Mr Trump and his advisors was" your views on Islam are ignorant".
"We've shown in London that there's nothing incompatible with being a mainstream Muslim and Western liberal values, and we showed that comprehensively on 5 May," he added.
Mr Trump, who is the presumptive nominee for the Republican Party after pushing out more than a dozen rival presidential candidates during the US primary season, reiterated that he backed the UK leaving the European Union.
In contrast to US President Barack Obama, who has warned that an EU exit would leave the UK at the "back of the queue" in trade talks, Mr Trump said he did not think it would harm the UK's trade position.
"It wouldn't make any difference to me whether they were in the EU or not," he said. "They certainly wouldn't be back of the queue, that I can tell you."
Scottish Conservative leader Ruth Davidson told ITV's Good Morning Britain that it "can't be good" if Mr Trump were to be elected as US president in November.
"I'd agree with that. I have huge and infinite faith in the American people that he won't be," she said.