Speculation is swirling over the op-ed's author, with top Trump administration officials - ranging from Vice President Mike Pence to Defense Secretary James Mattis - have issued statements denying their involvement.
Still Trump's call is the latest test of the independence of the Justice Department, which is supposed to make investigative and charging decisions without political interference from the White House.
"I would say Jeff should be investigating who the author of that piece was because I really believe it's national security", Trump said aboard Air Force One.
A "whodunnit?" style guessing game has raged in the corridors of power and on social media over the identity of the author, prompting almost every cabinet-ranked member of the government to deny involvement.
"What this person did is badly hurt the effort to rein in Trump ... and it will make him insane", said a Republican close to the White House. "The @nytimes should be ashamed and so should the person who wrote the false, illogical, and gutless op-ed", Pence's spokesman Jarrod Agen wrote on Twitter.
Trump lashed out at the author and at the "dishonest" Times.
James Dao, who runs the Times op-ed page, told CNN he received the article several days ago through an intermediary, but had spoken directly with its author.
"I think it's a disgrace that somebody can do that and I think it's more disgraceful that The New York Times would do it, but that somebody is allowed to do that, it's a very sad commentary", he said.
Mr Pompeo, who was in India, denied writing the anonymous opinion piece, saying: "It's not mine".
In response, Donald Trump has called the writer "gutless" and demanded the newspaper "turn him or her over to government at once!"
The writer of the Times op-ed said Trump aides were aware of the president's faults and "many of the senior officials in his own administration are working diligently from within to frustrate parts of his agenda and his worst inclinations".
The manifesto followed the release of excerpts from a bombshell book by Watergate reporter Bob Woodward, who portrayed Trump's White House as an out-of-control "crazytown". Some people who agreed with the writer's points suggested the president's reaction actually confirmed the author's concerns.
"It may be cold comfort in this chaotic era, but Americans should know that there are adults in the room". "We fully recognize what is happening".
"I'm looking at that right now".
It continued on: "The President's threats both underscore why we must safeguard the identity of the writer of this Op-Ed and serve as a reminder of the importance of a free and independent press to American democracy".
In one eye-opening passage, the writer says cabinet members initially considered invoking the 25th Amendment of the US Constitution, which provides for the president's removal if he is "unable to discharge the powers and duties of his office". "They are the only ones complicit in this deceitful act".
He accused the media of trying to undermine the Trump administration, and said he found that "incredibly disturbing".
Miller reported from Washington and Lemire reported from NY.