In private, the efforts to charm Mr. Trump were more pointed, according to a person familiar with the episode: Ms. Noem greeted him with a four-foot replica of Mount Rushmore that included a fifth presidential likeness: his.

But less than three weeks later, Ms. Noem came to the White House with far less fanfare — to meet not with Mr. Trump, but with Vice President Mike Pence. Word had circulated through the Trump administration that she was ingratiating herself with the president, fueling suspicions that there might have been a discussion about her serving as his running mate in November. Ms. Noem assured Mr. Pence that she wanted to help the ticket however she could, according to an official present.

She never stated it directly, but the vice president found her message clear: She was not after his job…

“There seems like there might be some interest on her part — it certainly gets noticed,” Jon Hansen, a Republican state representative in South Dakota, said of Ms. Noem’s positioning for national office.