A high ranking high ranking member of the executive branch reportedly was serious about being added. According to Fox News:
“[Trump] said, ‘Kristi, come on over here. Shake my hand,’” Noem said, according to the newspaper.
“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 thought the president was joking, the paper reported.
“I started laughing,” she said. “He wasn't laughing, so he was totally serious.”
No one from the White House responded when the Argus Leader checked to verify the story. But the newspaper noted that Trump did mention Mount Rushmore during a rally in Youngstown, Ohio, last July.
This answer wouldn't be complete without the relevant visual aid (image from the Daily Mail):

Unfortunately, the people involved do not comment on the process of adding another person tot the carving. Regardless of that, given these rumours by high-ranking officials and the lack of denial from the White House, this might be the start of a process (even if it's not carved in stone yet).