Then, he called it a "good idea."
"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 said in a tweet late Sunday.
He also posted a photo from this year's July Fourth weekend campaign-style appearance at Mount Rushmore -- with his face carefully aligned with those of the other four of the nation's most memorable presidents: George Washington, Thomas Jefferson, Theodore Roosevelt and Abraham Lincoln.
The New York Times reported earlier Sunday that when Noem greeted Trump upon his arrival in South Dakota on July 3, she presented him with a four-foot replica of Mount Rushmore that included his face.
According to the Times, a Republican official familiar with the matter said a White House aide reached out to Noem's office last year to ask the process to add additional presidents to the monument.
Noem, a rising star in the Republican Party, knew even before then that Trump is keen on the idea.
In a 2018 interview with the Sioux Falls Argus Leader, Noem said the two struck up a conversation about the prospect in the Oval Office during their first meeting.
"I started laughing," she said. "He wasn't laughing, so he was totally serious."
She continued, "He said, 'Kristi, come on over here. Shake my hand, and so 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?'"
Trump also suggested he be added to Mount Rushmore at a campaign rally in Youngstown, Ohio, in 2017.
“I’d ask whether you think someday I might 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,'" Trump said.