President Donald Trump on Friday offered good wishes to Britain's former ambassador to Washington Kim Darroch, who resigned earlier this week amid a spat with the Republican president following leaked diplomatic cables.
The statement also said there has been damage caused to Britain's global relations, and there would be clear public interest in bringing the person or people responsible to justice.
In a highly unusual move, he also warned the media that they could face prosecution if there was any further publication of the documents.
British lawmakers are warning that the country is at risk of turning into a "police state", as police tell media outlets not to publish leaked government documents after a row that led to the resignation of the UK's ambassador to the US.
"Once you start prosecuting the media for publishing stuff you don't like, that is a very unsafe place to be", Bob Seely, a Tory MP who sits on the Commons Foreign Affairs committee, told the Telegraph.
John Whittingdale, the Tory MP and former culture secretary, said: "The idea of prosecuting journalists is completely wrong".
"Going after the leaker is fair game, going after the media is not", he said.
"I doubt it is a crime to publish".
The Counter Terrorism Command is responsible for investigating breaches of the Official Secrets Act. That is the law of the land.
Sir Kim announced on Wednesday he was resigning, saying his position had become "impossible" following the leak of his dispatches in which he described Donald Trump's White House as "inept" and "dysfunctional".
In his Twitter storm on Tuesday, Trump described the ambassador as "wacky", "a very stupid guy" and a "pompous fool".
Almost a week after the cables were leaked, Basu's announcement of the investigation ended with a pointed call to the media. You can stop this now.
Meanwhile, Mr Trump has suggested he was too quick to denounce Sir Kim following the leak.
Trump earlier had publicly blasted the former British diplomat after reports unveiled details of Darroch's telegrams assessing the US government and Trump's administration. "He said some things that were pretty nice from the British ambassador".
"He was given a perfectly easy opportunity to say: "I'm sorry about this, he's an excellent ambassador and has my support", which we ought to give him because he is undoubtedly a superb ambassador and a very good civil servant, and he chose quite deliberately not to".
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