The ambassador’s comments are said to have been made in May 2018 after then-foreign secretary Boris Johnson made a failed trip to the White House in a bid to change Mr Trump’s mind on leaving the Joint Comprehensive Plan of Action.
The revelation came as police identified a suspect behind the leak, according to The Sunday Times.
Just hours earlier, Mr Johnson and Tory leadership rival Jeremy Hunt criticised Metropolitan Police assistant commissioner Neil Basu for warning journalists they could face prosecution for publishing the memos.
Mr Basu said the leaked emails could be a “criminal matter” that was not in the public interest, and that a police investigation had been launched into a potential breach of the Official Secrets Act.
Mr Hunt said he would “defend to the hilt the right of the press to publish those leaks if they receive them and judge them to be in the public interest”.
And Mr Johnson said prosecution “would amount to an infringement on press freedom and have a chilling effect on public debate”.
Shadow foreign secretary Emily Thornberry told BBC1's Andrew Marr Show: "It was wrong for it to be leaked, it was wrong for President Trump to throw a wobbly and behave like a toddler, it was wrong for Theresa May to be as weak as she was.
"But the real villain of the piece is Boris Johnson, who refused to stand with one of our most senior diplomats. That's not the way to behave."
And work and pensions secretary Amber Rudd said: "I was disappointed by Boris Johnson's response. I think the correct response was to stand up for our ambassador."
In a memo to Downing Street on 8 May last year, sent after Mr Johnson returned to London, Sir Kim said the Trump administration was “set upon an act of diplomatic vandalism, seemingly for ideological and personality reasons – it was Obama’s deal”, The Mail on Sunday reported.
He is said to have suggested that there were splits among Mr Trump’s closest advisers – with Mike Pompeo, the secretary of state, distancing himself from the president’s actions – and that the White House lacked a strategy on what to do following its withdrawal from the Iran deal.
Neither Mr Pompeo nor Mike Pence, the vice president, or John Bolton, the national security adviser, “could articulate why the president was determined to withdraw, beyond his campaign promises”, Sir Kim added.
Under the terms of the agreement – still supported by Britain, France and Germany – international sanctions on Iran were eased in return for Tehran accepting curbs on its nuclear programme.
In a second cable sent later that day, the former ambassador reportedly wrote that “following a typically hyperbolic statement on the nature of the ‘murderous’ Iranian regime, Mr Trump signed a presidential memorandum to start the process of reinstating US nuclear sanctions”.
Sir Kim resigned from his role on Wednesday after leaked documents published a week ago by The Mail on Sunday revealed he had described Mr Trump’s administration as “dysfunctional” and “inept”.
The president lashed out at the former ambassador in a string of tweets, saying Washington would “no longer deal” with him, and that he was “not liked or well thought of” within the country.
Sir Kim quit his post, saying his job had become “impossible” after Mr Trump’s tirade against him.
It is believed he made the decision after Tuesday’s televised Tory leadership debate, during which Mr Johnson refused to rule out replacing him.