Donald Trump has called on the attorney general, Jeff Sessions, to investigate the New York Times after it published an opinion piece by an anonymous senior administration official which detailed a “quiet resistance” working to rein in the president and raised questions about his fitness for office.

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Speaking to reporters onboard Air Force One on Friday, Trump said the column should be investigated as a national security matter and that he was weighing taking action against the news outlet.

“We’re going to see. I’m looking at that right now,” Trump said. Sessions “should be investigating who the author of that piece was because I really believe it’s national security”.

Trump has accused the New York Times of committing “treason” after the paper published the article on Wednesday, which claimed that a coalition was at work within the administration to thwart Trump’s “worst inclinations”.

More than two dozen Trump administration officials have denied being the author, amid fevered speculation over the identity of the writer. Vice-president Mike Pence, the secretary of state, Mike Pompeo, and the US ambassador to the United Nations, Nikki Haley, were among a cast of senior officials and cabinet members to publicly reject responsibility for the anonymous account on Thursday.

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The first lady, Melania Trump, even published a statement after suggestions that she could have written it, calling on the author to come forward and say publicly what he or she was willing to write anonymously.

“If a person is bold enough to accuse people of negative actions, they have a responsibility to publicly stand by their words and people have the right to be able to defend themselves,” she said. “To the writer of the op-ed: you are not protecting this country, you are sabotaging it with your cowardly actions.”

In his first major political speech since leaving the White House, former president Barack Obama denounced Trump’s response to the article, as part of a wider rebuke of his successor’s behaviour and agenda.

“It shouldn’t be Democratic or Republican to say that we don’t threaten the freedom of the press because they say things or publish stories we don’t like,” Obama said.

He also criticised the claim made in the article which asserted that the coalition of “resistance” within the Trump administration was working in the interests of the American people.

“The claim that everything will turn out OK because there are people inside the White House who secretly aren’t following the president’s orders? That is not a check … That is not how our democracy is supposed to work … These people aren’t accountable,” he said.

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During a campaign appearance in Billings, Montana, on Thursday night, Trump doubled down on his attacks against the Times, telling a raucous crowd of supporters: “Nobody knows who the hell he is, or she, but for the sake of our national security, the New York Times should publish his name at once.

“Unelected, deep-state operatives who defy the voters to push their own secret agendas are truly a threat to democracy itself.”

Earlier in the day, the White House aggressively responded to the column as speculation mounted over the author’s identity. Sarah Sanders, the White House press secretary, tweeted the phone number of the New York Times opinion desk in an effort to amplify pressure on the paper to identify the “anonymous coward”.

During an interview with Fox & Friends, conducted onstage before Trump’s rally and set to air on Friday, the president called the paper’s decision to publish the column “very unfair”.

The editorial was published as the White House was contending with yet another firestorm. A book authored by the celebrated journalist Bob Woodward, poised for release next week, chronicles the chaos and dysfunction within the Trump administration.

Excerpts released on Tuesday provided an unflattering portrait of the president, who was described by aides in disparaging terms that included being likened to a schoolchild.

Woodward, who is most famous for breaking the story of the Watergate scandal that prompted the resignation of Richard Nixon from the presidency in 1974, wrote that White House chief of staff John Kelly was so incensed by Trump’s behavior that he privately described the president to other aides as an “idiot” and complained that they were in “Crazytown”.

Sanders derided Fear as “nothing more than fabricated stories, many by former disgruntled employees, told to make the president look bad”.