US voters will reject Donald Trump because his "ignorant, divisive and dangerous" views play into the hands of extremists, Sadiq Khan believes.
The public spat between the new Mayor of London and the presumptive Republican presidential nominee resumed after Mr Trump called the prominent Muslim politician "ignorant" and "nasty" in an interview with ITV's Good Morning Britain.
A spokesman for the mayor hit back, saying: "Donald Trump's views are ignorant, divisive and dangerous - it's the politics of fear at its worst and will be rejected at the ballot box just as it was in London.
"Sadiq has spent his whole life fighting extremism, but Trump's remarks make that fight much harder for us all - it plays straight into the extremists' hands and makes both our countries less safe."
He said there were "no plans" to seek direct talks and mocked Mr Trump's challenge to the mayor to take an IQ test.
"Ignorance is not the same thing as lack of intelligence," he said.
Mr Trump's campaign call for a temporary ban on Muslims entering the US led to almost universal condemnation from UK politicians - including British Prime Minister David Cameron, who called it "divisive, stupid and wrong".
Asked about potential future trans-Atlantic tensions if he becomes president, the ultra-wealthy tycoon told GMB: "It looks like we are not going to have a very good relationship.
"Who knows, I hope to have a good relationship with him but he's not willing to address the problem either."
Number 10 said the PM "has made his views clear".
Mr Trump had appeared to offer an olive branch to Mr Khan after his election to City Hall - in the face of a Conservative campaign critics compared with the Republican's tone - saying there would "always be exemptions" to the ban.
But he said he would "remember" the hostile reaction that he received from the mayor, who said his own election had shown voters would not back "divisive" candidates.
"He doesn't know me, hasn't met me, doesn't know what I'm all about. I think they were very rude statements and, frankly, tell him I will remember those statements. They are very nasty statements," he told GMB.
"When he won I wished him well. Now, I don't care about him, I mean, it doesn't make any difference to me, let's see how he does, let's see if he's a good mayor."
Mr Trump said he was offended by Mr Khan's public denouncement but denied he was "at war" with him.
"I just think it's very rude of him. In fact it's the opposite. I wished him well when I heard he won, he's a Muslim, I think it's ignorant for him to say that."
The presidential hopeful said the policies he mooted on the campaign trail were just "suggestions", but said there was a "tremendous" problem with Islamic extremism.
He said: "It's not Sweden doing the damage - we have a real problem and we have to discuss it."