The threat provoked widespread anger on social media and elsewhere with many attacking the president for directly challenging the First Amendment, which guarantees the freedom of the press.
The president was reacting to an NBC News report that he pushed senior aides for a major expansion of the USA nuclear arsenal.
Trump said he has never discussed increasing the number of nuclear weapons the USA has.
The tweets were then posted on Wednesday morning.
"No, I never discussed increasing it".
MSNBC reported in 2016 that as a candidate, Trump asked a foreign policy adviser three times in a one-hour meeting why the United States could not deploy its nuclear weapons.
The Executive Director of the First Amendment Coalition, David Snyder, told Newsweek the comment from Trump was something more akin to a statement "the leader of some authoritarian country would make". I want it in ideal shape.
CNN has until recently been the president's favorite target for accusations of broadcasting 'fake news'.
During the meeting, the president was shown a slide that depicted the decrease in US nuclear weapons that started in the late 1960s, the news station reported.
Such a move would abrogate key global arms control treaties and upend decades of non-proliferation policy, with deep ramifications across the globe. I want it in ideal shape. Two people familiar with the discussion said the meeting was unproductive to the point that advisers made a decision to continue the talks the next day at the Pentagon with a smaller group.
As president and candidate, Trump has often mused aloud about America's nuclear weapons power, saying there is no point in having them if you do not use them.
Broadcast licenses have previously been revoked by authoritarian leaders including the late former president of Venezuela Hugo Chavez, who revoked a number of broadcast licenses of stations that displayed stringent anti-Chavez views.
In reality, a challenge to NBC's broadcast licences would be a lengthy process involving petitions at the FCC.
The Trump administration has threatened to prosecute leakers.