He slammed The New York Times and CNN -- the publications which first reported on the alleged request -- saying that he "never suggested" anything similar.
Still, Trump argued that he deserves a spot on the landmark because of his administration's many accomplishments.
"This is Fake News by the failing @nytimes & bad ratings @CNN. 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!"Reports that Trump wants his own bust alongside Theodore Roosevelt, Abraham Lincoln, George Washington and Thomas Jefferson surfaced earlier this week.
During his first meeting with Republican South Dakota Gov. Kristi Noem, Trump reportedly raised the idea of having his image carved into Mount Rushmore.
"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," Noem recalled in an interview.
After the meeting, a White House aide reportedly reached out to Noem, asking if additional presidents can be added to the mountain sculpture.
However, it is not possible to add another face to the landmark, since the only remaining space on it is thought to be too unstable.
Reports about the White House reaching out to Noem's office came amid rumors that Trump could replace Vice President Mike Pence with the South Dakota governor. Noem has rejected the idea, saying that she would not want to replace Pence on the Republican 2020 ticket.
Trump has long expressed admiration for Mount Rushmore, vowing to protect the landmark from those who want to destroy it.
On Sunday, Trump also tweeted a photo of himself next to the monument.Trump visited the monument on Fourth of July, when he delivered a controversial speech.
During the address, the commander in chief took aim at the "radical left" and news media. He accused the media of "slandering" the military forces and the American people as a whole.
Trump also discussed Black Lives Matter protesters' demands to remove Confederate statues. He delivered the remarks amid nationwide protests against police brutality and racial inequality, which began following the May 25 death of George Floyd.