"There is no more carvable space up on the sculpture," a Mount Rushmore spokeswoman has said

President Donald Trump at Mount Rushmore
| Credit: SAUL LOEB/AFP via Getty

While President Donald Trump this weekend denied a New York Times report that said his aides asked Mount Rushmore officials about adding additional presidents to the monument, he tweeted that it nonetheless "sounds like a good idea to me."

"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, 74, tweeted.

He has floated such an idea before. But Mount Rushmore officials have said it's not possible to add more figures to the face of the South Dakota mountain in the Black Hills (even if there was a groundswell of support for adding Trump in the future).

"There is no more carvable space up on the sculpture," a spokeswoman at Mount Rushmore told The Sioux Falls Argus Leader in 2018, when South Dakota Gov. Kristi Noem said that Trump had talked about his aspirations to be on the mountain.

"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," the spokeswoman explained to the paper then.

"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," she said.

From left: President Donald Trump and First Lady Melania Trump at Mount Rushmore
| Credit: SAUL LOEB/AFP via Getty

Trump's monumental ambitions — to be represented alongside former Presidents George Washington, Abraham Lincoln, Theodore Roosevelt and Thomas Jefferson — came up again this weekend, following the Times article on how Noem reportedly attempted to woo the president during his July 3 visit for Independence Day.

The Times reported that Noem had presented Trump with a 4-foot model of the Mount Rushmore carving that added his likeness.

What's more, according to the paper, White House aides had previously reached out to the governor's office asking about the process to add more faces to Mount Rushmore.

Another Trump official denied the rumor to the Times, and the president similarly tweeted that it "never happened." (The White House did not respond to PEOPLE's request for comment.)

The president's desire to be on the monument is no secret though.

Noem revealed in a 2018 interview with former Survivor contestant Mitchell Olsen that Trump had once asked her about adding his face to Mount Rushmore during the first meeting they ever had in the Oval Office.

"He said, 'Kristi, come on over here. Shake my hand,' " the governor said. "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 said she thought Trump was joking and so she started to laugh, but then she realized "he was totally serious."

Mount Rushmore
Mount Rushmore
| Credit: SAUL LOEB/AFP via Getty

During a speech in Ohio in 2017, the president said, “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!’ So I won’t say it, okay? I won’t say.”

The director of the South Dakota State Historical Society told The Washington Post that same year that his “best gut feeling is that there will never be a change to the memorial.”