Reached for comment, a Justice Department spokesperson told CNBC that the department "does not confirm or deny the existence or nonexistence of investigations."

"We're going to take a look at what he had, what he gave, what he's talking about, also where he is right now," Trump said of the unnamed individual, who has not been identified as either a man or a woman. If that person has a high-level security clearance, Trump said, "I don't want him in meetings" on sensitive national security issues.

Asked by a journalist whether Sessions should investigate the source of the critical column, Trump said, "I think so. Because I think it's national security — I would say Jeff should be investigating who the author of that piece was because I really believe it's national security."

President Donald Trump on Friday said he wants Attorney General Jeff Sessions to investigate the identity of the author of an anonymous op-ed that was published in The New York Times on Wednesday.

Trump also said he was looking into potentially taking legal action against The New York Times, citing the vague national security concerns he mentioned beforehand.

The president said it was a "disgrace" for the newspaper to have published the op-ed from a senior administration official. "For somebody to do this is very low, and I think, journalistically and from many different standpoints, and maybe even from the standpoint of national security, we'll find out about that," he said. The New York Times' stock dipped slightly following Trump's comments, but quickly recovered.

In a statement Friday afternoon, the Times said it was, "confident the Department of Justice understands that the First Amendment protects all American citizens and that it would not participate in such a blatant abuse of government power."

This is not the first time Trump has mentioned that he thinks there could be a national security angle attached to the publication of the op-ed, which described a secret effort underway inside the administration to "frustrate parts of his agenda and his worst inclinations."

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Trump also marveled at the bipartisan nature of the criticism directed at the author of the op-ed in the past day. "So many people that never said a good thing about me are now saying that [resistance from inside the government] should never happen, [and] have actually got to my side," he said.

Indeed, at precisely the same moment that Trump was talking to reporters on Air Force One, a few hundred miles east in Illinois, former President Barack Obama was speaking at a campaign event, where he agreed with the president. "That's not how our democracy is supposed to work," Obama said, with "people inside who secretly aren't following the president's orders. That is not a check."

Obama continued: "These people aren't elected. They're not accountable. They're not doing us a service by actively promoting 90 percent of the crazy stuff that's coming out of this White House and saying, 'Don't worry, we're preventing the other 10 percent.' That's not how things are supposed to work."

On Wednesday, the highest branches of the U.S. government were rocked by the op-ed, which described what its author called "the work of the steady state," as opposed to the "deep state."

The op-ed also described "early whispers within the cabinet of invoking the 25th Amendment, which would start a complex process for removing the president. But no one wanted to precipitate a constitutional crisis," the author wrote. "So we will do what we can to steer the administration in the right direction until — one way or another — it's over."

In the 24 hours following the op-ed's publication on Wednesday, more than 20 top Trump administration officials, including nearly every member of the president's Cabinet, issued formal statements denying authorship of the column.

— CNBC's Kevin Breuninger contributed to this story.