President Donald Trump tweeted Monday that he thought it would be a "good idea," to have his face ono Mount Rushmore but making any additions to the landmark – whether it would be to add the 45th president or any other person – is likely not possible without jeopardize the existing structure.

Trump, who has floated the idea in the past of having his face on landmark, posted the tweet in response to a New York Times report that the White House reached out to South Dakota Gov. Kristi Noem to inquire about what would be involved in making the addition.

"This is Fake News by the failing @nytimes & bad ratings @CNN," the president tweeted Sunday night, linking to a CNN article about the Times report. "Never suggested it although, based on all of the many things accomplished during the first 3 1/2 years, perhaps more than any other Presidency, sounds like a good idea to me!"

That tweet immediately followed one featuring a photo of Trump standing before the Black Hills monument from a perspective that put his face in line with those of former Presidents George Washington, Thomas Jefferson, Theodore Roosevelt and Abraham Lincoln.

In response to Trump's denial, New York Times spokesperson Ari Isaacman Bevacqua told USA TODAY, "We stand by our reporting."