Sadiq Khan has hit back at “preposterous” claims by Donald Trump after the US president criticised the London mayor for doing a “a bad job” on terrorism and crime in the capital.

In an interview in the Sun, Trump lashed out at Khan, who has allowed a Trump protest balloon to float above Westminster during his visit.

The US president reignited his feud with Khan, who criticised the US president as “ill-informed” over a tweet following the London Bridge attack in June 2017.

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“I think allowing millions and millions of people to come into Europe is very, very sad ... You have a mayor who has done a terrible job in London,” Trump said.

“He has done a terrible job. Take a look at the terrorism that is taking place. Look at what is going on in London. I think he has done a very bad job on terrorism.

“I think he has done a bad job on crime, if you look, all of the horrible things going on there, with all of the crime that is being brought in.”

Khan responded on Friday morning, saying terrorism was a global problem, with people dying all over Europe. “What is interesting is Trump is not criticising mayors of those cities, but he is criticising me,” he said in an interview on the BBC’s Today programme.

Trump’s comments blaming immigration for crime in England were “preposterous”, he added.

“There has been an increase in violent crime across England and Wales … it’s gone up by more than 20% … and by 4% in London ... We must deal with the causes, but also enforcement and where we have lost £7m in our budget in London I have invested £4m ... [but] the idea to blame immigration from Africa is preposterous, and we should call him out when does,” Khan said.

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Trump’s verbal attack on Khan came after the London mayor refused to block a plan to fly a giant inflatable “Trump baby” near parliament to coincide with the president’s visit to the UK. The 20ft-high blimp depicts the US leader as an angry infant wearing a nappy and clutching a mobile phone.

Khan defended the decision, saying to have blocked the inflatable would have been to inhibit freedom of speech.

“The UK, like the USA, has a long and rich history of rights and the freedom to protest and freedom of speech. The US ambassador himself commented that one thing [the] USA and the UK have in common is freedom of speech, and the idea of restricting that and [the] right to assemble because someone is offended by something is [a] slippery slope,” he said.

“When determining these things [it should be about] ... whether it is safe and peaceful. As a politician I should not be the arbiter of what is good or bad taste.”

Khan said there would be far-right and pro-Trump demonstrations taking place on Saturday. “I disagree with their views as well, but letting them protest should not mean endorsing their views ... I am not banning the pro-Trump demo either,” he added.