"I'm going to be photographed with every possible leader", he insisted, and later posed with the president for an official welcome alongside North Atlantic Treaty Organisation chief Jens Stoltenberg.
But McDonnell insisted Jeremy Corbyn would work with the U.S. president.
Trump fuelled the row on his state visit to Britain earlier this year by stating that "everything" would be on the table in any post-Brexit trade talks.
Fact checkers have said that if the document is legitimate, it shows the United States is interested in the UK's health and pharmaceutical markets, but says nothing about where the UK stands on it.
"Like Boris Johnson, Donald Trump is hardly known for his honesty". The British public are "firmly against" an NHS sell-off.
Johnson was filmed in conversation with Canada's Justin Trudeau, French President Emmanuel Macron and Dutch Prime Minister Mark Rutte at Buckingham Palace late on Tuesday.
In a speech, Mr Johnson will tell the other North Atlantic Treaty Organisation leaders: "Seventy years on, we are rock solid in our commitment to North Atlantic Treaty Organisation and to the giant shield of solidarity that now protects 29 countries and almost a billion people".
McDonnell's words echo former Conservative Party chairman Lord Patten's warning yesterday that Donald Trump and Boris Johnson share an "elastic relationship with the truth."
But today Donald Trump said he wouldn't want to service even if it was "handed on a silver platter" as he arrived in the United Kingdom for a two day North Atlantic Treaty Organisation summit.
Trump said: "We have absolutely nothing to do with it, and we wouldn't want to".
"I believe in one instance he claimed he didn't know what the initials NHS stood for".
Johnson insisted the North Atlantic Treaty Organisation meeting had been "extremely practical and extremely harmonious".
Johnson has said that "under no circumstances" would the NHS be on the table in negotiations.
Britain's decision on whether to allow Huawei a role in building 5G telecoms networks will be based on ensuring the national security and ensuring cooperation with the USA -led Five Eyes intelligence alliance, Prime Minister Boris Johnson said on Wednesday.
"Boris Johnson must give a guarantee that our farmers and world-leading food standards will not be sacrificed on the altar of a Trump trade deal".
The Tories said Labour's claim it would reduce living costs "defies belief", while accusing the party of standing on a manifesto containing tax hikes for "ordinary hardworking people".
Meanwhile on the campaign trail, Labour vowed to end "rip-off Britain" as party research alleges the Tories have cost families almost £6,000 a year by failing to curb rising bills.