Some of their signs read "Dump Trump", "Lock Him Up" and "There Will Be Hell Toupee". Any contravention, it warns, "could result in force being used against you, your arrest and/or prosecution".
However, the president also suggested that the dispute with Khan was personal and that the mayor had shown disrespect to him by allowing a 20-foot "Trump Baby" balloon - an orange likeness of Trump wearing a diaper - to fly over London protests during the president's visit.
"With the flexibility that is built into the white paper proposals, there is absolutely no reason why a trade deal between us and the USA can not happen and I think both sides do want it to happen".
"My message to those coming to the protests in London is that this must be peaceful and good-spirited", he said.
Foreign Office Minister Sir Alan Duncan said he did not think Mr Trump's remarks were rude.
He had been a regular visitor to the country for many years, but this will be his first trip since becoming president.
On reaching the residence the pair walked across the lawn hand-in-hand.
But they avoided the noise and spectacle of the protests on the ground by travelling by helicopter between the airport, London and Oxfordshire.
Despite the welcome from May, many Britons are opposed to Trump's visit.
Before the Sun interview was published, May invoked Winston Churchill as she addressed Trump and business leaders at a black-tie dinner at Blenheim Palace, the grand 18th century country house that was the birthplace of the British World War Two leader.
Trump arrived at the reception to much pomp and ceremony, with the bands of the Scots, Irish and Welsh Guards playing in his honour. The performance included a bagpiper playing Amazing Grace.
Trump may not be able to ignore the presence of protesters entirely.
The US embassy's alert to American citizens was an unusual step, CNN reported. "Be aware of your surroundings (and) exercise caution if unexpectedly in the vicinity of large gatherings that may become violent", it added.
The president and first lady will then head to Windsor Castle to meet the Queen before leaving for Scotland in the evening.
The demonstrations come ahead of a mass protest scheduled to begin at 2:00pm on Friday in central London, and dozens of other actions against Trump-amounting to what the Stop Trump Coalition and Stand Up to Trump have dubbed a "carnival of resistance".
Claiming Khan has done a "very bad job on terrorism", as well as on crime. The president also said North Atlantic Treaty Organisation had "never been more united", a day after he roiled the military alliance's annual summit in Belgium by insulting allies' defense spending. But on Thursday they said they welcomed the plan and would study it closely.
Vickers added: "He would be wise not to attempt to kiss her, and I don't expect for a moment that he will".
"Melania has seen photos of the Trump baby blimp, and she doesn't find it amusing in the slightest, she thinks it's mortifying", a source close to the first lady tells HollywoodLife EXCLUSIVELY. "I think he's got what it takes". Trump said that would happen over "a fairly short period of time".
"But what has to happen first.is that we reach an agreement with the European Union itself, so that we can exactly piece together how we can do the trade deals we want to do with the rest of the world". Its key provision is a "customs arrangement" with the EU that would see the United Kingdom remain largely within the bloc's customs union.
On Saturday, Trump supporters will rally near the USA embassy in Nine Elms, south London, for an event titled "Welcoming Trump to Britain Rally - Make Britain Great Again", at which a number of right-wing figures are scheduled to speak.