Trump first raised the prospect of having his face carved on the historic site soon after he took office, in January 2017. Kristi Noem, who at the time was a Congresswoman representing South Dakota, said he mentioned the idea during that first meeting. 'He said: "Kristi, come on over here. Shake my hand,"' Noem recalled. '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 he was joking. 'I started laughing,' she said. 'He wasn't laughing, so he was totally serious.'
Later that year, Trump raised it again - this time in public, at a rally in Youngstown, Ohio, in July. '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,' he said. '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. A website has been set up advocating for Obama. McGee-Ballinger said it was impossible. 'There is no more carvable space up on the sculpture,' she said.
https://www.dailymail.co.uk/news/article-8610013/White-House-reached-South-Dakota-governor-adding-Trump-Mount-Rushmore.html