"I would say Jeff [Sessions] should be investigating who the author of that piece was because I really believe it's national security".
"I think if we are going to meet, it has to be a fair meeting", Trump said.
Reporting to be from anonymous senior U.S. government official, the story described the USA president as adrift from reality and unstable.
But Mr 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.
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. The former Air Force translator had worked as a contractor at a National Security Agency office in Augusta, Georgia, when she printed a classified report and left the building with it hidden in her pantyhose.
"The Times should never have done that, because really what they've done is virtually, you know, it's treason", the president said.More news: Saturday Schedule: Serena Williams faces Naomi Osaka for history
However, he said: "If you thought elections don't matter, I hope these past two years have corrected that impression".
"A lot of people have said they didn't do, nearly all of the high-level people have said that".
While crediting the few successes Trump's White House has seen, the essay adds that "these successes have come despite - not because of - the president's leadership style, which is impetuous, adversarial, petty and ineffective".
Trump has gone as far as to call the letter treasonous in his first tweet critical of the piece.
In the CBS interview, Pence blasted the op-ed and the Times, saying that "the author of the anonymous editorial, and frankly The New York Times, should be ashamed".
The book depicts Trump aides working to head off potential foreign policy disasters, disregarding a call by the president for the assassination of his Syrian counterpart, and removing from his desk an order that would have canceled a US-South Korea trade agreement. Victor told the Federal Bureau of Investigation she mailed the document to an online news outlet.
President Donald Trump's reaction was a combination of rage and self-pity.
"Our libel laws should be toughened up so you can't do that", he said.
"Many of the senior officials in his (Trump's) own administration are working diligently from within to frustrate parts of his agenda and his worst inclinations", the author writes. "And we are trying to do what's right even when Donald Trump won't".