Trump has lambasted Khan on Twitter, accusing him of making a "pathetic excuse", for saying Londoners should not be alarmed by the sight of additional police on the streets of the British capital after Saturday's attack that killed seven people.
"The invitation has been issued and accepted and I see no reason to change that, but as far as what Sadiq Khan has said about the reassurances he's offered the people of London, I think he was entirely right to speak in the way he did," Johnson said in a BBC radio interview when asked whether Trump's state visit should be canceled.
No date has been set for the visit, which was agreed during Prime Minister Theresa May's visit to Washington in January.
"I don't wish to enter into a row between those two individuals who are I think are probably perfectly able to stick up for themselves," Johnson said of Trump and Khan.
May has said Khan is doing a good job, echoing public sentiment across London.
Khan, the first Muslim elected as London's mayor, and Trump have been at odds since Khan was strongly critical of Trump's election campaign pledge to impose a temporary ban on Muslims entering the United States. Since taking office, Trump has ordered temporary travel restrictions on people from several Muslim-majority countries, although the ban is currently held up by federal courts.
Asked if he would like Trump's visit to be called off, in an interview with Channel 4 News on Monday evening, Khan said his position toward Trump remained the same.
"I don't think we should roll out the red carpet to the president of the USA in the circumstances where his policies go against everything we stand for," Khan said.
Former Democratic U.S. presidential candidate Hillary Clinton, defeated by Trump last November, praised Khan's performance in dealing with the London attacks.
"It is time for steady, determined leadership like we are seeing from London's mayor and local authorities," Clinton said at a fundraising event on Monday, according to The Washington Examiner. She did not mention Trump by name, but she said it was "not the time to lash out, to incite fear and use trash talk and terror for political gain."
Deputy White House spokeswoman Sarah Sanders said on Monday that she did not think it was correct to characterize Trump's tweets as "picking a fight" with Khan.
Asked whether Trump was attacking the mayor because he is Muslim, Sanders told reporters: "Not at all. And I think to suggest something like that is utterly ridiculous."
Trump's oldest son, Donald Trump Jr., defended his father.
"Every time he puts something out there he gets criticized by the media. All day, every day," Trump Jr. said in an interview with ABC's "Good Morning America" broadcast on Tuesday.
"And guess what, he's been proven right about it, every time. We keep saying, 'It's going to be great' and 'Hold fast,' 'We're going to keep calm and carry on.' Maybe we have to keep calm and actually do something," he said. He was referring to a slogan of resilience from World War Two, to "keep calm and carry on", that Britons have echoed following the London attacks.
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