For much of the past year, Donald J. Trump has described his perfect running mate as someone who could easily take over as commander in chief if needed and help him draw a contrast to President Biden and Vice President Kamala Harris.

“It’s got to be somebody that can be a good president, which Biden doesn’t have,” Mr. Trump said in April during an interview with Hugh Hewitt, a conservative political commentator.

But now, as the curtain closes on his theatrical selection process, his public statements suggest that his thinking has shifted and he’s giving more weight to political calculations.

Mr. Trump said on Monday his top priority remained someone qualified to be president and then quickly added a second criterion: “Somebody that helps you get elected.”

His new focus on someone who can help him win may stem from indecision over his choice, some allies said. Mr. Trump has said he plans to announce his running mate before or during the Republican National Convention, which begins Monday. Some aides have wondered whether he will announce his pick before then, in time for a campaign rally on Saturday in western Pennsylvania.

Two people who spoke separately with Mr. Trump this week said they had the impression he was still undecided. As of Friday morning, he had not called any of his top three contenders to offer them the position, according to two advisers who insisted on anonymity to discuss internal deliberations.

Mr. Trump said six months ago that he had settled on a pick, but he also has a history of vacillating over important decisions. One example was his previous running-mate selection.

In 2016, he didn’t announce his selection of Mike Pence until two days before the Republican National Convention. The day before their first news conference together, Mr. Trump spent hours second-guessing his choice and asking aides if there was still time to choose former Gov. Chris Christie of New Jersey or the former House Speaker Newt Gingrich of Georgia.

Mr. Trump’s earlier statements about seeking a running mate who “can be a good president” grew from discussions with advisers about how voters this year are so intensely familiar with both Mr. Trump and Mr. Biden. In a race with such well-known nominees, the thinking went, there would be little need for a running mate to serve as a bridge to a particular political constituency or a specific battleground state.

This year, Mr. Trump’s final three contenders are Senator J.D. Vance of Ohio, Gov. Doug Burgum of North Dakota and Senator Marco Rubio of Florida. If Mr. Trump is still considering their political attributes, each would bring a different set of advantages and disadvantages.

Mr. Vance would be the most likely to energize Mr. Trump’s conservative base, but also may be the richest target for Democratic attacks. Mr. Burgum, a wealthy two-term governor, could reassure pro-business Republicans but is less tested on a national stage. Mr. Rubio has proved his appeal among Latino voters, but his ability to perform under pressure is viewed skeptically in some corners of the party.

Just a few years ago, Mr. Vance was among Mr. Trump’s most biting critics, and he left a trail of video clips for Democrats to use against the former president. Even as a relatively new convert to the MAGA movement, Mr. Vance carries a controversial record, including his support for a national abortion ban. He has tried to temper that message to better fit with Mr. Trump’s concerns that abortion politics are hurting Republicans electorally, even as Mr. Vance’s website pledges to “end abortion.”

Mr. Vance, 39, has the least amount of political experience on his résumé, after being sworn into his first public office just last year. That could undercut Mr. Trump’s attacks on Ms. Harris as being ill-prepared to step in for Mr. Biden if necessary.

“J.D. has a body of anti-Trump statements that can be exploited,” said David Axelrod, a senior strategist on President Barack Obama’s presidential campaigns. “He can be painted in the most extreme corners of his party, and his preparedness is a very legitimate question.”

But Mr. Vance’s hardscrabble upbringing could appeal to Rust Belt voters who may decide the election, and his allies view his youth as an asset in a race that features an 81-year-old incumbent and a 78-year-old opponent. Mr. Vance would be the only person at the top or bottom of the ticket with military service. And even his detractors privately acknowledge that he often comes across as articulate and compelling on television, an issue of outsize importance to Mr. Trump.

“I think J.D. Vance would be the greatest threat to Kamala Harris” in a potential vice-presidential debate, Ashley Etienne, a former communications director for Ms. Harris, said on CNN last month.

A former venture capitalist and author, Mr. Vance has been the top choice for some of the leading voices within Mr. Trump’s political base, many of whom are combative commentators, including Dan Bongino, a former New York City police officer turned pundit; Tucker Carlson, the former Fox News host; and Mr. Trump’s eldest son, Donald Trump Jr.

Donald Trump Jr. has been passionate about his father’s selection, using his podcast this week to critique some of his father’s top contenders. He diminished Mr. Burgum as a better fit to run the Energy Department and struck some deeply conspiratorial notes in describing Mr. Rubio as a potential impeachment risk for his father.

He suggested Senate Republicans would be more likely to remove Mr. Trump from office if it meant Mr. Rubio would take over, hinting that establishment Republicans in the chamber would work against his anti-establishment father. Such a scenario ignores that Senate Republicans voted twice to acquit Mr. Trump in impeachment trials.

Mr. Rubio has the longest relationship with Mr. Trump, which dates back to their caustic presidential primary rivalry in 2016 in which they repeatedly traded insults. Mr. Rubio has struggled in some of his biggest moments, including a bungled response to Mr. Obama’s State of the Union in 2013, and has had to reassure conservatives after his role that year in a bipartisan immigration bill.

Still, the two men resolved their differences and became confidantes while serving in Washington. Mr. Rubio is the son of Cuban immigrants — his father was a bartender and his mother, a factory worker — and his personal story could hold some appeal for Mr. Trump’s blue-collar supporters.

Mr. Rubio is the only Republican elected to three Senate terms in Florida, a state that has receded as a top presidential battleground. On this electoral canvas, Mr. Rubio has performed well with Hispanic voters in the southeast corner of the state and with more conservative white voters in the Panhandle. In 2016, more Floridians voted in the general election for Mr. Rubio’s Senate bid than Mr. Trump’s presidential ticket.

“Marco Rubio would be a great vice-presidential pick,” Representative Carlos Gimenez, a Florida Republican, said Wednesday on Fox Business. “He’ll bring more Hispanics to the Republican Party, he will add to the base and I think he would make a great vice president.”

Mr. Burgum, a two-term governor from North Dakota, is the most recent arrival in Mr. Trump’s circle after his little-noticed presidential bid last year caught the former president’s attention.

“I actually like the governor; I respect him a lot,” Mr. Trump told Newsmax after the first Republican presidential debate in August. “He’s got something very good about him — he’s a high-quality person.”

But Mr. Trump has expressed some concern about Mr. Burgum’s signature on one of the most restrictive abortion laws in the nation, even though Mr. Burgum has pledged to oppose a national ban on the procedure. Including him on the ticket could help settle pro-business Republicans who are nervous about Mr. Trump’s unpredictability, but he remains largely unknown — and relatively untested — on a national stage.

Still, the wealthy former business executive has become a tireless advocate for Mr. Trump on cable news and on the campaign trail, earning support from some key conservative figures. On Thursday, the editorial boards of The Wall Street Journal and The New York Post, each part of the media empire controlled by Rupert Murdoch, the conservative billionaire, backed Mr. Burgum as the best choice for Mr. Trump.

Pete Hoekstra, the Republican Party chairman in Michigan, a crucial battleground state, lobbied for Mr. Burgum this week as a running mate who could “help win Michigan and other key Midwest states.”

“He’s a problem solver,” Mr. Hoekstra said in a social media post, “and has Midwest common sense.”

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