The anonymous New York Times column published Wednesday described a culture of "quiet resistance" toward the president within the White House, with aides subtly ignoring or blocking his decisions when they believe him to be acting dangerously.
He also attributed the op-ed to the "deep state", writing on Twitter, "The Deep State and the Left, and their vehicle, the Fake News Media, are going insane - & they don't know what to do".
Invoking national security concerns, Trump said the paper should publish the official's name, before encouraging reporters to investigate the matter.
In an interview Thursday with Fox News, Trump said it was unfair for the person to pen the editorial anonymously because there's no way to discredit it.
On the Internet and in Washington, a guessing game has raged over who the author might be, prompting almost every cabinet-ranked member of the government to deny it was them.
The op-ed's publication fueled speculation about its source, and on Thursday morning a stream of senior administration officials came forward to deny that they had written the column, among them Vice President Pence, Secretary of State Mike Pompeo and Defense Secretary Jim Mattis.
As the New York Post noted, the op-ed used the term "lodestar", a seldom-used word that is a favorite of Vice President Mike Pence. The @nytimes should be ashamed and so should the person who wrote the false, illogical, and gutless op-ed. His deputy chief of staff said Pence's office is "above such amateur acts". White House press secretary Sarah Huckabee Sanders called on the anonymous official to resign.More news: Ayanna Pressley wins Democratic primary- will become first Black congresswoman in MA
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In the op-ed, the writer said that "ours is not the popular "resistance" of the left", and cited "bright spots" of Trump's tenure, including deregulation, tax reform and increased military spending.
"I come from a place where if you're not in a position to execute the commander's intent, you have a singular option, that is to leave", he said.
James Dao, the newspaper's op-ed editor, said in the Times' daily podcast that while an intermediary brought him together with the author, he conducted a background check and spoke to the person to the point that he was "totally confident" in the identity.
Meanwhile Trump's United Nations envoy Nikki Haley was asked on her way to a meeting of the Security Council if she was the anonymous correspondent and responded, simply: "No".
"We considered using the process spelled out in the constitution but decided that would be too messy and opted for trying a slow-motion coup instead, but then we made a decision to undermine that by going public about it anonymously".
Trump called Woodward's new work "fiction".
The Times opinion section said the piece was written by a senior official in the administration and that it was taking the rare step of publishing an anonymous article because disclosing the author's identity would jeopardise the person's job.