The US President reacted by lashing out in a series of tweets, calling Darroch a "wacky Ambassador" and "very stupid guy".
The claims come after Scotland Yard was at the centre of a furious row over press freedom after warning journalists not to publish any more leaked cables from Britain's United States ambassador Sir Kim Darroch.
Darroch resigned after days of scathing criticism from the U.S. president on Twitter.
The report comes after Darroch's resignation in light of leaked diplomatic cables in which he called Trump "inept" and "incompetent", and called the Trump White House "uniquely dysfunctional".
In this photo, taken on Friday, Oct. 20, 2017, British Ambassador Kim Darroch hosts a National Economists Club event at the British Embassy in Washington.
"It's not news that the USA and UK differ in how to ensure Iran is never able to acquire a nuclear weapon; but this does underline that we do not shy away from talking about our differences and working together", a spokesman for the UK Foreign Office told CNN when asked about the latest leaked cable.
The spokesman added: "It's not news that the USA and United Kingdom differ in how to ensure Iran is never able to acquire a nuclear weapon; but this does underline that we do not shy away from talking about our differences and working together".
Announcing the police inquiry, Counterterrorism police unit leader Neil Basu warned against any further publication of leaked documents.
Contentiously, police issued a warning to journalists that publishing the documents "could also constitute a criminal offence".
"I defend to the hilt the right of the press to publish those leaks if they receive them & judge them to be in the public interest: that is their job", he said.
Brexit party chairman angling for role?
The UK's Mail on Sunday newspaper reported that Kim Darroch wrote made the comments in a May 2018 memo, when he was still serving as the ambassador. According to the Sunday Times, news of the secret relationship between the pair is fueling the belief of civil servants and pro-EU lawmakers that Darroch "was brought down by conspirators keen to replace him with a pro-Brexit businessman'".
"Turn them over to the police or give them back to their rightful owner, Her Majesty's Government", Basu said.
Trump also said that the USA would "no longer deal with him".
At the time the USA withdrew from the deal, senior Trump administration officials - including Secretary of State Mike Pompeo and the Director of National Intelligence Dan Coats said Iran was adhering to its commitments under the deal.
Confrontations between Washington and Tehran have escalated, culminating in a plan for USA air strikes on Iran last month that Mr Trump called off at the last minute.
Trump left the 2015 Iran nuclear deal in 2018, claiming that the deal, which saw Iran agree to limit its sensitive nuclear activities in exchange for lifted economic sanctions, did not go far enough.
"Iran has always been secretly 'enriching, ' in total violation of the awful 150 Billion Dollar deal made by John Kerry and the Obama Administration", Trump wrote.
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