Governor of South Dakota Kristi Noem (R) recounted her first meeting with Trump in 2018 at the Oval office:

“He said, ‘Kristi, come on over here. Shake my hand. 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.”

According to The New York Times, a Presidential aide reached out to her later to inquire about the process for being added to Mount Rushmore. Trump immediately derided it as “fake news,” because immediately saying “sounds like a good idea to me!”

Trump brought up the idea of being put on Mount Rushmore in a July 2017 rally, though he said he was “totally joking.”

CNN – New York Times: White House reached out to South Dakota governor about adding Trump to Mount Rushmore
ABC News – Trump denies WH asked about adding him to Mount Rushmore, then calls it a ‘good idea’
Argus Leader – Mount Trumpmore? It’s the president’s ‘dream,’ Rep. Kristi Noem says