Former White House national security adviser John Bolton is under investigation by the Justice Department after publishing an explosive book that angered President Donald Trump, The New York Times reported on Tuesday.

The Times said the department had convened a grand jury to examine whether Bolton illegally published classified information in his book released in June.

It said publisher Simon & Schuster had been subpoenaed for communications relating to the book, "The Room Where it Happened”, which depicted Bolton’s former boss as reckless and corrupt, and supported the charges on which Trump was impeached in December.

After the White House failed to block the book’s publication, Trump demanded Bolton, a hawkish veteran diplomat, be investigated.

"Bolton broke the law and has been called out and rebuked for so doing, with a really big price to pay,” Trump tweeted on June 20.

"Washed up Creepster John Bolton is a lowlife who should be in jail, money seized, for disseminating, for profit, highly classified information,” he wrote three days later.

The Justice Department and Simon & Schuster declined to comment. Bolton’s attorney Charles Cooper said they were aware of the subpoena reports.

"Ambassador Bolton emphatically rejects any claim that he acted improperly, let alone criminally, in connection with the publication of his book,” Cooper said in a statement.

NPR public radio also reported the investigation, adding that Bolton’s literary agent had also been subpoenaed.

A case against Bolton would focus on his claim that his manuscript had passed through a pre-publication national security review and claims by critics that it did not complete that review.

Cooper said before publication that it had undergone a line-by-line review and was cleared by a senior White House national security official.

The judge that rejected the White House push to block the book — partly on grounds that copies were already widely distributed — warned that Bolton had "gambled with the national security” of the country and exposed himself to civil and criminal liability.

Around the same time Paul Nakasone, the director of the National Security Agency, said he had "identified classified information” in the manuscript in a pre-publication review.

"Compromise of this information could result in the permanent loss of a valuable SIGINT [signals intelligence] source and cause irreparable damage” to US intelligence, Nakasone said, without specifying which information.

Director of National Intelligence John Ratcliffe also suggested in June that the book contained secrets.

"The type of classified information in these passages is the type of information that foreign adversaries of the United States seek to obtain, at great cost, through covert intelligence collection,” he said.

The Times said the Justice Department debated whether to open an investigation of Bolton with some arguing that Trump’s comments in June would make it appear like political retribution.