Assistant Commissioner of the Metropolitan Police Neil Basu speaks to the media after a auto crashed outside the Houses of Parliament in Westminster, London, Britain, August 14, 2018.
Osborne described the statement as "ill-advised".
Trump also said that the U.S. would "no longer deal with him".
He said, "I would advise all owners, editors and publishers of social and mainstream media not to publish leaked government documents that may already be in their possession, or which may be offered to them, and to turn them over to the police or give them back to their rightful owner, Her Majesty's Government".
Peter Spiegel, Financial Times US managing editor, wrote: "Well, this is rather chilling from a major police force in a western democracy".
Norman Lamb, the former health minister who now heads the science and technology committee as a Liberal Democratic MP, said: "We can't contemplate any slippery slope to a police state that accepts that sort of limitation on the freedom of the press to report".
The Foreign Secretary said he would "defend to the hilt" the right of the press to publish leaks they judge to be in the public interest.
London Police said its terrorism command had initiated the probe into the leaking.
He said: "A prosecution on this basis would amount to an infringement on press freedom and have a chilling effect on public debate".
A Scotland Yard investigation into a breach of the Official Secrets Act is underway.
"They should also be aware of the huge damage that has already been done, and the potentially even greater damage to be done by further breaches of the Official Secrets Act".
British police say they have opened an investigation into a leak of confidential memos that led to the resignation of the UK's ambassador to Washington.
The leaked memos caused an global row, with the United States president responding to the revelation of Sir Kim's memos by branding the diplomat "wacky" and "a very stupid guy", as well as vowing to "no longer deal with him".
Shortly afterwards Sir Kim announced his resignation saying "the current situation is making it impossible for me to carry out my role".
Meanwhile, Mr Trump has suggested he was too quick to denounce Sir Kim following the leak.
However Mr Johnson has sought to distance himself from speculation that his comments caused Sir Kim to quit, saying "I think that unfortunately what I said on that TV debate was misrepresented to Kim".
Speaking to reporters at the White House, Mr Trump said he had since been told Sir Kim had said "some very good things" about him.
It comes as Boris Johnson, front runner to be Britain's next prime minister, denied he was responsible for the resignation of Darroch but admitted his comments had been a factor in the shock departure of one of the country's most senior diplomats.
The new prime minister will be named on July 23 and take office the following day.
Turkey drills off Cyprus in bid for regional influence
The Yavuz drillship dropped anchor on Monday to the south of Cyprus' Karpasia peninsula, a jutting northeastern panhandle. In 1974, following a coup aiming at Cyprus's annexation by Greece, Ankara intervened as a guarantor power.
How Arsenal Transfer News Affect Their Odds to Win?
Other players rumored to be destined for Arsenal include; Sokratis Papastathopoulos, from Borussia Dortmund, and Saul Niguez. If they are rumored to buy an elite player, during the transfer window, they play well which increases the odds of winning.