Memos from Darroch describing the Trump administration as inept were leaked to a British newspaper, infuriating the USA president, who launched a Twitter attack on both the envoy and outgoing prime minister Theresa May, who had given Darroch her full support.
According to Britain's Mail on Sunday newspaper, which reported on Darroch's memos, the envoy wrote there was a "credible path" for Trump winning a second term.
It was the first time Mr Trump has spoken publicly about the subject since Sir Kim resigned earlier this week after his diplomatic cables, some of which portrayed the Trump administration in an unflattering light, were leaked.
"The investigation will be reviewed at every stage to ensure a proportionate investigation is undertaken".
Met Police Assistant Commissioner Neil Basu said: "Given the widely reported consequences of that leak, I am satisfied that there has been damage caused to United Kingdom global relations, and there would be clear public interest in bringing the person or people responsible to justice".More news: SC fixes petition to investigate Judge Malik video for hearing
In his Twitter storm on Tuesday, Trump described the ambassador as "wacky", "a very stupid guy" and a "pompous fool".
Mr Johnson denied any culpability, saying he had spoken to Sir Kim on Thursday to express his sadness over his resignation and the ambassador told him he had not watched the ITV debate.
Boris Johnson (L) refused to back British ambassador Kim Darroch (R) during a televised debate.
"I would say to the person or people who did this, the impact of what you have done is obvious. Turn yourself in at the earliest opportunity, explain yourself and face the consequences".
"The publication of leaked communications, knowing the damage they have caused or are likely to cause may also be a criminal matter", the statement from Basu said.More news: Donald Trump tears into 'lame duck failure' Paul Ryan
But on Friday, Mr Johnson told the BBC a "misrepresented" account of his remarks relayed to Sir Kim had been "a factor" in his decision to step down.
The British police has also advised "all owners, editors and publishers of social and mainstream media" not to publish leaked government documents, whether they already have them or are offered to publish new ones.
He replied: "Well I think it would have been better if the whole subject had never come up".
"Some people just told me - too bad - they said he actually said some very good things about me", he said.
At one point he also goaded Hunt, asking how long he would keep Darroch in place after vowing to retain him.More news: 4.6 natural disaster rattles Seattle region in the middle of the night