The New York Times Opinion desk acknowledged the extraordinary step of publishing such an anonymous op-ed, saying it did so at the request of the author whose identity is known to the paper.
Former CIA Director John Brennan, a fierce Trump critic, called the op-ed "active insubordination. born out of loyalty to the country".
"The idea I ever called the President an idiot is not true", Kelly wrote, "in fact it's exactly the opposite".
The piece immediately became the top story in the country after it was published online by the Times Wednesday afternoon, dominating the news cycle and unleashing a fury of gossip.
He said the scoop from the op-ed is essentially "Trump's not nice".
A spokesman for US Vice President Mike Pence's office also criticised the newspaper and said Mr Pence does not write anonymous opinion columns.
It was Trump's latest broadside against author Bob Woodward's new book, Fear: Trump in the White House, excerpts of which first appeared Tuesday, with countless anecdotes painting Trump as dangerously ignorant of world affairs and his White House devastatingly beset by internal feuds.
A senior official told Vanity Fair that Trump "hates" the leaks and was particularly upset about reports that Cohn intentionally "stole a letter off Trump's desk" that would have removed the US from the Korea Free Trade Agreement.
The column, released Wednesday, was said to be written by an anonymous senior administration official working from "within" the White House in order to "thwart" U.S. President Donald Trump's "worst inclinations".
A "very small number of people within the Times who know this person's identity", Dao reportedly said, adding that the company had spoken with the senior official directly.
Jodi Kantor, an investigative reporter with the New York Times, tweeted: 'So basically: Times reporters now must try to unearth the identity of an author that our colleagues in Opinion have sworn to protect with anonymity? "I don't know and neither do you, but it doesn't seem likely".
CNN's Kate Bennett asked FLOTUS how she felt about the op-ed, which is making headlines nationally, and she was hardly complimentary of its author.
The author wrote that "there were early whispers within the Cabinet of invoking the 25th Amendment, which would start a complex process for removing the president".
There have been previous reports by journalists and former White House staffers of upheaval in the White House West Wing since Trump's January 2017 inauguration.
"We believe publishing this essay anonymously is the only way to deliver an important perspective to our readers", the newspaper said.
U.S. President Donald Trump holds a meeting with Republican House and Senate leadership in the Roosevelt Room at the White House in Washington, D.C., U.S. September 5, 2018. "It would have to clearly depend upon who was writing this, and the amount of trust and confidence that we had in going ahead", Ancrum said.
The White House press secretary, Sarah Huckabee Sanders, called on the "coward" who wrote the piece to "do the right thing and resign".