Regardless, the mayor suggested that "name-calling" should be beneath any world leader, and expressed surprise that Trump would "behave like an 11-year-old".
The manuscript, known as the Sussex Declaration, is one of only two known handwritten parchment copies of America's formative text.
Earlier, Trump was pictured giving the Queen an unusual handshake as they met at the start of his state visit.
He spoke by phone with former Foreign Minister and former London mayor Boris Johnson and said on Tuesday he thought Johnson "would do a good job".
President Trump doubled down on his criticisms of London Mayor Sadiq Khan during a press conference alongside outgoing British Prime Minister Theresa May, a jab that Karl Rove said he was better off avoiding.
May will make clear to Trump that Britain is reviewing "the right policy" on Chinese company Huawei and its involvement in the UK's 5G telecommunications network, the spokeswoman added.
Be it projecting influence for peace, standing ready to fight, or delivering vital aid around the world, HMS Queen Elizabeth and her sister ship HMS Prince of Wales, will fly the flag of a Global Britain.
Impersonator Phil Ball said the US President was "not an ethical person" who would look to take advantage of the United Kingdom in any future trade deal between the two nations post-Brexit.
At a joint news conference with May, Trump predicted Brexit would happen. "I think there's a great opportunity to greatly enlarge that, especially now", he said. May is stepping down as head of her party at the end of the week, entering a lame duck period as the conservatives select a new leader.
New tariffs against Mexico will begin to bite next week, President Trump vowed Tuesday, unless the White House is satisfied that Mexico's government is acting with new alacrity to stop migrants from crossing into the United States.
The second day of Trump's state visit will be filled with more official business than ceremony, he meets with May and business leaders to discuss trade and other matters.
Trump, when asked whether changing British food standards and the publicly funded National Health Service might be included in a deal, said that "everything will be on the table, absolutely".
Late-night hosts Stephen Colbert and Trevor Noah took a swipe at Donald Trump once again Monday. Trump told the Sunday Times in an interview that Britain should "walk away" from talks and refuse to pay a $49 billion United States divorce bill if it doesn't get better terms from the EU.
Also featured in Trafalgar Square: a 16-foot-tall, talking, animated "robot" of a plastic-cast Trump character, sitting on a golden toilet, his red tie dangling into bowl, tweeting.
Hundreds of thousands protested against him during a trip a year ago and a blimp depicting Trump as a snarling, nappy-clad baby will fly outside Britain's parliament during the visit.
A giant inflatable blimp depicting Mr Trump as a pouting baby in a diaper was raised outside the British parliament in London on Tuesday ahead of what is expected to be one of the city's largest protests against a foreign leader.