The 45th US president made the remarks on Twitter where he posted a picture of himself next to the iconic monument. Mr Trump has openly floated the idea before in 2018, and last year a White House aide is said to have asked if there is space for another President on Mount Rushmore.
Mr Trump replied to a New York Times article about the White House aides inquiry into adding a president to the site.
He savaged the article in a post, by denying he ever requested to be added.
Mr Trump also branded CNN and the Times as “fake news” for publishing the claims.
But he added: “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!”
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It’s not the only time Mr Trump has made the request, with the first time coming shortly after his inauguration in 2017.
Ms Noem recalled that Mr Trump inquired about adding his face to Mount Rushmore in their first meeting.
She 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?’“
“I started laughing. He wasn’t laughing, so he was totally serious.”
Mr Trump raised the concept again in an Ohio rally the same year.
He added: “’I’d ask whether or not you think I will someday be on Mount Rushmore, but 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, OK? I won’t say it.”
Maureen McGee-Ballinger, public information officer at Mount Rushmore, told The Argus Leader that workers are asked daily whether any president can be added, with many people suggesting Franklin Roosevelt, John F. Kennedy and Ronald Reagan, among others.
A website was also set up advocating for Barack Obama to be added.
But in a major blow to Mr Trump’s dream, Ms McGee-Ballinger said it was impossible.
She said: “There is no more carvable space up on the sculpture.”