Donald J. Trump’s resilience in polls of the Republican presidential primary field is shifting attention to what, for the moment at least, is the only truly competitive national race for 2024: the contest to be his running mate.
Speculation over Mr. Trump’s potential vice president — a decision that would rest solely with him — has remained an undercurrent in the primary race as his rivals for the nomination, including former Gov. Nikki Haley and Senator Tim Scott, a pair of South Carolina Republicans, regularly distance themselves from questions about their possible interest in the No. 2 job.
One Republican welcoming those questions has been Gov. Kristi Noem of South Dakota. She isn’t running for the White House, and she recently told Fox News that “of course” she would consider joining a ticket with Mr. Trump.
Ms. Noem, 51, fueled further vice-presidential conjecture by endorsing the former president at a rally on Friday evening in her home state.
“Tonight, Mr. President, my message is clear: It is an honor to have you with us in South Dakota,” Ms. Noem said in front of a crowd of thousands in Rapid City. “You made America great again once. Let’s do it again.”
Mr. Trump took the stage, briefly hugging Ms. Noem and exchanging words. Then, for a quick second, a graphic reading “TRUMP NOEM 2024” flashed on the giant screen above the stage.
“Kristi is a warrior for American values,” Mr. Trump said, going on to compliment her handling of the coronavirus pandemic and her policies in South Dakota.
“I get endorsements, some good, some bad,” he said. “Some don’t mean anything. Hers means a lot.”
While South Dakota holds little sway in the Republican presidential primary contest — and even less in a general election — Ms. Noem’s endorsement is noteworthy, because only eight of the nation’s 26 Republican governors have publicly picked sides so far.
Beyond Ms. Noem, just three — Mike Dunleavy of Alaska, Jim Justice of West Virginia and Henry McMaster of South Carolina — have backed Mr. Trump. Two others — Ron DeSantis of Florida and Doug Burgum of North Dakota — are running against him.
“Everybody should consider it,” Ms. Noem told Fox News about a potential vice-presidential slot. “If President Trump is going to be back in the White House, I’d do all I can to help him be successful.”
Still, Mr. Trump is said to be giving little direct thought to a running mate.
Some close to the former president said that was most likely rooted in superstition that such consideration would jeopardize his own nomination. Others said he had devalued the position, viewing it as little more than a White House staff position that carries little political sway with voters.
Mr. Trump raised eyebrows among some associates with private, offhand comments that Mr. Scott had not received much coverage for his performance during the first Republican presidential debate. Mr. Scott has been mentioned as a potential vice-presidential pick even though he is currently running against Mr. Trump, who didn’t participate in his party’s first debate.
Steven Cheung, a spokesman for Mr. Trump, said the vice-presidential speculation showed that “everyone knows President Trump will be the nominee and he continues to dominate every single poll.”
Along with Mr. Scott, other Republican candidates mentioned as potential running mates for Mr. Trump have included Ms. Haley and the businessman Vivek Ramaswamy. Mr. Trump’s two-time running mate, former Vice President Mike Pence, has split with the former president over the 2020 election results. This week, he cast Mr. Trump’s populism as “a road to ruin” for the party.
In a radio interview this week, Mr. Trump told Hugh Hewitt, a conservative talk show host, that he was unlikely to make an early decision on a vice president — brushing aside the idea that his running mate could help campaign next spring when the former president is facing multiple criminal trials.
“There’s never been a vice president that got a president elected, because it doesn’t work that way,” Mr. Trump said. “It sounds good and everything, but the president gets himself elected.”
Mr. Trump endorsed Ms. Noem for governor in 2018, and she was an ardent ally during his presidency. When she hosted him in 2020, her laudatory public remarks prompted speculation that she was hoping to replace Mr. Pence on the Republican ticket.
Ms. Noem changed her tune somewhat after Republicans fell short of expectations in last year’s midterm elections. In an interview with The New York Times at the time, Ms. Noem — who was frequently cited as a potential 2024 candidate — floated the thought that she did not believe Mr. Trump offered “the best chance” for Republicans.
Still, Ms. Noem stayed out of a crowded Republican primary in which Mr. Trump is far and away the front-runner, and she has more recently voiced support for him in cable news appearances.
On Thursday, she told the conservative news channel Newsmax that she would “in a heartbeat” consider being Mr. Trump’s running mate if asked.
Maggie Haberman contributed reporting.
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