At the North Atlantic Treaty Organisation headquarters, Mr Trump spoke about his relations with the UK.
Following the departure of senior figures, including foreign minister Boris Johnson and Brexit minister David Davis, the political atmosphere is febrile. The protocol is that if embassy staff are warned internally about possible threats, then the American public must be as well, she said.
Mrs May said: "Our long term objective remains a constructive relationship with Moscow. It's not for me to decide what's in good taste or bad taste".
Protests are expected against Trump, whose policies - including a travel ban on a number of predominantly Muslim countries, detention of child migrants on the US-Mexico border and imposition of tariffs on European Union steel and aluminium exports - have all been criticised by the UK.
As well as in London, protests are expected to take place in Bristol, Newcastle, Leeds, Cambridge and Cardiff after Trump arrives here on Thursday.
On Thursday, Trump will travel to Blenheim Palace, the 18th-century mansion where Britain's World War Two leader Winston Churchill was born and spent most of his childhood. From there his meetings with Mrs May will be at her country home, Chequers, which offers a "more informal setting", according to a Downing Street spokesman.
Next, on July 13, a demonstration has been organised by the "Stop Trump Coalition" with the initial protest movement.
At the same time, Trump will be touring a defense site with May outside of London.
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United States ambassador Woody Johnson insisted the balloon was not a fair representation of British views.
The Republicans Overseas UK group expects up to 100 people to attend its event on Friday evening to show there are people who are happy about the trip, chairwoman Sarah Elliott said.
Most of the protests will take place that day in central London, the USA officials said, with some occurring the days before and after.
More than 50,000 people have signed up to demonstrate in London, where they intend to fly an oversized balloon depicting the president as an angry baby in a diaper.
But the scale of protest comes as no surprise and is unlikely to annoy the president, Ms Elliott said. "Are you honestly saying that the right to protest, the right to assemble, the right to free speech is limited by the ridicule it might cause to someone else?"
Mr Morgan, the forthright journalist and broadcaster, hit back and questioned why the London mayor had not been so vocal over the visit of controversial Turkish President Recep Erdogan.