Lib Dem leader Tim Farron wants Donald Trump's UK visit called off
Mr Trump, during a series of tweets following the London Bridge terror assault, misrepresented Mr Khan's statement that there was "no reason to be alarmed" when seeking to reassure Londoners.
This prompted Mr Khan's spokesman to state the mayor has "more important things to do than respond to Donald Trump's ill-informed" tweet, before Mr Khan later suggested he does not regard Mr Trump's planned state visit as welcome.
Mr Farron tweeted: “Theresa May must withdraw the state visit. This is a man insulting our national values at a time of introspection and mourning.
“After three terrorist attacks, @realDonaldTrump uses his time in the Oval Office to attack @MayorofLondon. He is an embarrassment to America.”
Sadiq Khan has been criticised by Donald Trump
This is a man insulting our national values at a time of introspection and mourning.
Mr Trump sparked outrage by continuing to criticise Mr Khan, tweeting: "Pathetic excuse by London Mayor Sadiq Khan who had to think fast on his 'no reason to be alarmed' statement.
"MSM (mainstream media) is working hard to sell it!"
The president's initial criticism of Mr Khan was based on a clear misinterpretation of some of the mayor's comments in the wake of the atrocity.
Speaking on the BBC, Mr Khan made clear that he was advising people not to be concerned by the presence of additional armed police officers on the streets of the capital in the days to come, and not playing down anxieties about the attack itself.
Mrs May initially sidestepped questions about Mr Trump by choosing to praise Mr Khan's response.
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And she stopped short of directly criticising the US president.
Asked if Mr Trump was wrong in his assessment of the London mayor, the Prime Minister told reporters: "I think Sadiq Khan is doing a good job and it's wrong to say anything else – he's doing a good job."
Mrs May, taking questions after a central London election campaign speech, was earlier asked if she wanted a period of silence from Mr Trump.
She replied: "I'm very clear that Sadiq is doing a good job as mayor of London.
"We're working with him, working together and that's important – central government and the London mayoralty and his officials working together to ensure we are responding to the attack and looking, as I said earlier, at the work that the police is doing to give the public extra protection and extra reassurance.
London Mayor Sadiq Khan and Met Police chief Cressida Dick
"We want people to go about their business."
Mr Khan appeared to criticise Mr Trump while addressing police funding following a visit to London Bridge.
He noted London has other responsibilities due to its status as a capital city, adding: "From major events, there's a big cricket match today, to state visits – some welcome, some less so.
"But the fact is we need to make sure the police have the resources and support they need, and as the mayor, my job is to make our city safe and I'll do my best as the mayor to be an advocate and champion for our police to get the tools they need."