President Trump on Sunday denied a New York Times report that a White House aide had asked South Dakota’s governor about how to add another president to Mount Rushmore. But Trump also suggested he wouldn’t mind seeing his own face etched into the monument.

“This is Fake News,” Trump tweeted of 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!”

Trump has a long history of saying his likeness should be added to Mount Rushmore, although in public he’s generally insisted he’s joking.

His fixation with Mount Rushmore as a populist symbol, though, is undeniable. In July, he staged a gala Independence Day celebration at the South Dakota monument where he tried to capitalize on the social and political divisions riling the nation amid the novel coronavirus pandemic in a fiery speech warning of a “left-wing cultural revolution."

President Trump said he would protect South Dakota's Mount Rushmore during a speech on July 3. (Reuters)

In the lead-up to that event, the Times reported on Saturday, a White House aide had asked the office of South Dakota Gov. Kristi L. Noem (R) to spell out exactly what the process might look like to add another presidential face to the stone facade.

Per the Times, Noem responded in kind: By presenting the president with a four-foot-tall reproduction of Mount Rushmore that included Trump’s face etched in the rock next to George Washington, Thomas Jefferson, Theodore Roosevelt and Abraham Lincoln.

Although Trump on Sunday disputed the Times account, Noem has said in the past that Trump personally raised the question with her. In 2018, Noem, then a member of Congress running for governor, said Trump brought up the idea of adding his face to Rushmore during their first meeting in the Oval Office.

“He said, ‘Kristi, come on over here. Shake my hand,’ ” Noem said, according to the Argus Leader. “I shook his hand, and I said, ‘Mr. President, you should come to South Dakota sometime. We have Mount Rushmore.’ And he goes, ‘Do you know it’s my dream to have my face on Mount Rushmore?’ ”

Noem broke out in laughter, she told the paper — but quickly realized Trump wasn’t joking.

“I started laughing,” she said. “He wasn’t laughing, so he was totally serious.”

In 2017, Trump made similar comments at a speech in Ohio, but insisted they were in jest.

“I’d ask whether or not you think I will someday be on Mount Rushmore, but, no — here’s the problem. If I did it joking, totally joking, having fun, the fake-news media will say, ‘He believes he should be on Mount Rushmore!’ ” Trump said at the time. “So I won’t say it, okay? I won’t say.”

So would it be possible to add Trump’s face to the mountain? There’s no clear process to make it happen, The Washington Post’s Philip Bump reported in 2017, though Trump would probably have to start by getting South Dakota’s legislature and possibly Congress to sign on. Adding a new face would cost at least $64 million in labor alone, Bump calculated.

But there’s also the question of whether the rock would be stable enough to add another towering set of presidential features. Jefferson’s tribute had to be moved from its original spot due to flaws in the granite.

The staff at Mount Rushmore says any further sculpting is simply impossible.

“There is no more carvable space up on the sculpture,” Maureen McGee-Ballinger, a spokeswoman for the monument, told the Argus Leader two years ago. “When you are looking on the sculpture, it appears there might be some space on the left next to Washington or right next to Lincoln. You are either looking at the rock that is beyond the sculpture (on the right), which is an optical illusion, or on the left, that is not carvable.”

The White House didn’t immediately respond to a message from The Post about Trump’s latest reported flirtation with a spot in the rock.