Police now say there is a "clear public interest" in bringing the person or people responsible to justice, adding that the UK's global relations had been damaged by the incident.
The probe is being led by the Metropolitan Police Counter Terrorism Command because it is responsible for investigating breaches of Britain's Official Secrets Act.
In a statement, the London Metropolitan police's assistant commissioner, Neil Basu, said that "given the widely reported consequences of that leak", there had been "damage caused to United Kingdom global relations" and that there "would be clear public interest in bringing the person or people responsible to justice".
He noted that the investigation will be reviewed at every stage to ensure a proportionate investigation is undertaken.
Johnson pointedly refused to back Darroch during the televised debate on Tuesday, leading to accusations from fellow Conservative Party MPs that he had thrown the ambassador "under the bus" in order to bolster his own ties with Trump.
"From that moment on it became hard for Sir Kim to continue as ambassador".
In the emails leaked to the Mail on Sunday, Sir Kim said: "We don't really believe this administration is going to become substantially more normal; less dysfunctional; less unpredictable; less faction-riven; less diplomatically clumsy and inept".
Shadow global minister Liz McInnes said Sir Kim Darroch changed into once "factual doing his job" and the legal investigation changed into once "welcome".
Mr Basu continued: 'I would say to the person or people who did this, the impact of what you have done is obvious.
"However, you for the time being are additionally liable for diverting busy detectives from endeavor their core mission". You can stop this now.
Basu said that the leaks may be a "criminal matter" as he urged the culprit to "turn yourself in at the earliest opportunity, explain yourself and face the consequences".
Foreign Secretary Jeremy Hunt, who is jousting with Boris Johnson to become the next prime minister, tweeted Saturday that the person responsible for the leak must be found and held responsible, but he differed with police over whether the publication of leaks is a possible crime.
The Metropolitan police has asked anyone with information to come forward: "Exercise your public and civic duty and call the police on 0800 789321 without delay", the statement said. That call will be treated in strictest confidence.
However, he added: "We have also been told the publication of these specific documents, now knowing they may be a breach of the OSA [Official Secrets Act], could also constitute a criminal offence and one that carries no public interest defence".
His comments drew a furious tirade from the president who denounced him as a "very stupid guy" and a "pompous fool", and said the White House would no longer deal with him.
A first batch of leaked reports authored by Darroch last weekend caused major turmoil between Britain and its closest ally.