The president brought up his desire for a buildup during a meeting with top national security advisors in July, according to the report, which cited three officials at the gathering. Advisors told Trump about treaties that would be endangered and other hurdles preventing such a move. There is no planned expansion of nuclear weapons, NBC reported.
After the meeting ended, Secretary of State Rex Tillerson was heard calling Trump a "moron." That comment reportedly sparked more tensions between Tillerson and Trump following an NBC News report last week. That report also said Tillerson was close to resigning this summer. Tillerson subsequently denied that he had contemplated resigning but did not deny calling the president a "moron."
The report on the July meeting comes as the U.S. pushes for the denuclearization of North Korea.
Trump has made public statements before about boosting the U.S. nuclear arsenal. In December, he tweeted that the "United States must greatly strengthen and expand its nuclear capability until such time as the world comes to its senses regarding nukes."
Defense Secretary James Mattis in a statement Wednesday called the report "absolutely false" and "erroneous."
In a Wednesday morning tweet, the president claimed "fake" NBC "made up" the story. He called it "pure fiction, made up to demean."
In a second tweet, he suggested that the NBC coverage is "bad" for the country. He asked: "At what point is it appropriate to challenge their License?"
Later, the president continued his attacks over the critical coverage, telling reporters it is "frankly disgusting the way the press is able to write whatever they want to write."
Trump's tweet apparently references the licenses granted to individual television stations by the Federal Communications Commission. The FCC does not license news organizations or television networks.
NBC Universal, through its television stations division, owns 28 NBC and Telemundo local television stations.
The licensing of television stations dates to the early days of radio, when the government set regulations based on the idea that the spectrum belonged to the public. Radio and television stations are required to renew their licenses periodically, and those licenses can be revoked if the station's owner violates FCC regulations or other laws.
But as recently as December, the commission made clear that individual broadcasters have "broad discretion" in what they choose to air.
"The Commission will not take adverse action on a license renewal application based upon the subjective determination of a listener or group of listeners that the station has broadcast purportedly inappropriate programming," the FCC commissioners wrote in a recent decision challenging a local radio station license.
Disclosure: NBC and CNBC are owned by NBCUniversal, a division of Comcast.
— CNBC's John Schoen contributed to this report.