Ron DeSantis knows the statistics by heart.

He ticks them off as he contrasts his sweeping re-election as Florida governor with Republican losses nationwide last fall: a flip of traditionally “very blue” Miami-Dade County; the narrowness of his 2018 victory versus his landslide in 2022; the remarkable Republican voter registration gains in the state on his watch.

“There is no substitute for victory,” Mr. DeSantis said last week during his first trip to New Hampshire in his still-undeclared presidential bid. He denounced the “culture of losing” that he said had engulfed Republicans in recent years, swiping at Donald J. Trump in all but name.

“If the election of 2024 is a referendum on Joe Biden and his failed policies — and we provide a fresh vision for American renewal — Republicans will win the White House, the House and the U.S. Senate,” he told the crowd. “So we cannot get distracted, and we cannot afford to lose, because freedom is hanging in the balance.”

Electability has emerged as one of the early pressure points in the 2024 Republican presidential primary.

That amorphous, ill-defined, eye-of-the-beholder intangible — the sense of whether voters believe a politician can actually win — was supposed to be one of Mr. DeSantis’s strengths, tapping into the genuine Republican frustration with years of ballot box disappointments to urge a new face for the party in 2024. Republicans lost with Mr. Trump, the argument goes, but can win with Mr. DeSantis.

But there are growing questions about Mr. DeSantis’s own ability to win over the independent and suburban voters who delivered the White House to President Biden, and whether the hard-line stances the governor has taken, including on abortion, will repel the very voters he promises to win back. His feuding with Disney — including an offhand remark this week suggesting he would put a state prison next to Disney World — has raised alarms, even among would-be allies.

For years, electability has been the fool’s gold of Republican politics.

Since the rise of the Tea Party more than a decade ago, Republican primary voters have consistently cast ballots with their hearts, sneering at so-called experts to select uncompromising hard-liners as nominees. Even as losses in winnable races have mounted, the mere perception of running as electable has repeatedly backfired, giving off for many Republicans the stench of the reviled establishment.

“It has sounded like an excuse to get conservative voters to support somebody they don’t really want, even though the argument may very well be true,” said Whit Ayres, a veteran Republican pollster. Citing G.O.P. losses while Mr. Trump has defined the party — in 2018, 2020, 2021 and 2022 — Mr. Ayres added of the former president and the G.O.P. 2024 front-runner, “There is no education in the fifth kick of a mule, and yet it appears that’s where we’re headed.”

For Mr. Trump’s rivals, hitting him as an electoral loser is central to chiseling away at the crucial bloc of voters who liked his presidency but might be willing to move on. It also allows them to create contrast without directly crossing him; Nikki Haley, for instance, talks about the need for a “new generation” to win.

Core to Mr. DeSantis’s particular electability pitch is that he won in Florida despite not tacking to the middle: that voters, in other words, can have both a fighter and a winner.

But his recent signing of a six-week abortion ban puts him on the far right on an issue that Democrats have used to mobilize their base with great success since Roe v. Wade was overturned. And congressional Republicans, who have had a front-row seat to the party’s Trump-era struggles, have pointedly delivered far more endorsements to Mr. Trump, including from Mr. DeSantis’s home state delegation during his visit to Washington this week, in a sign of the governor’s slipping traction.

Mr. Trump’s team has pushed an electability case against Mr. DeSantis. A Trump-allied super PAC has run ads warning that Mr. DeSantis would go after Social Security and Medicare, touchstone issues that Democrats have used to defeat Republicans nationwide.

“If anyone thinks throwing seniors under the bus is a winning argument, they are seriously out of touch,” said Steven Cheung, a Trump spokesman. “There is only one electable candidate in 2024, and that is President Trump.”

The DeSantis team did not respond to a request for comment.

Sarah Longwell, a Republican who holds regular focus groups with G.O.P. voters, said in the immediate aftermath of the 2022 midterm losses, many Republicans had come to view Mr. Trump as an electoral loser.

“Baggage is the word you’d hear,” she said.

Mr. DeSantis was the beneficiary, rising as voters looked for a less polarizing alternative. “Trump with a mute button,” one voter memorably described a dream Republican candidate, she recalled.

That trend, however, has dissipated of late, said Ms. Longwell, who is involved with several groups that oppose Mr. Trump.

“The electability pitch really only works if there is lots and lots of polling showing Trump losing by a wide margin,” she said. In a 50-50 nation, Mr. Trump remains competitive with Mr. Biden in almost every public poll, even if Mr. DeSantis often performs marginally better.

Then there are the known unknowns of 2024 for Republican voters. If Mr. Trump loses the primary, would he sabotage the winner? And what would be the impact of further potential criminal indictments?

Vivek Ramaswamy, a former biotechnology executive running a long-shot Republican presidential campaign, said the belief that electability would carry the day in 2024 was “somewhere between a wish and a mirage.”

“It is fatally hubristic for anybody to think they can run that math equation,” Mr. Ramaswamy said in an interview aboard his campaign bus, adding that whoever was strongest would shift with debates and world events in the coming months.

Mr. Trump’s pundit-defying victory in 2016 has uniquely inoculated him from charges that he cannot win. And as Mr. Trump’s rivals in 2016 learned — like when Jeb Bush called him the “chaos candidate” — it can be especially hard to press a case about electability when trailing badly in the polls, as Mr. DeSantis does now.

In interviews, Trump supporters note that he only narrowly lost in 2020 despite a pandemic that crippled American life for months, circumstances that almost certainly won’t repeat. For all the turbulence Mr. Trump creates, they say he has been tested on the national stage in a way his opponents have not.

The who-can-win debate plays out strikingly differently between the two parties. In 2020, Democratic primary voters obsessed over electability before nominating Mr. Biden, who made his strength against Mr. Trump a centerpiece of his candidacy.

In New Hampshire, interviews with Republican voters, activists and party officials revealed both the fertile ground for and the challenges of any electability campaign against Mr. Trump. Mr. DeSantis arrived in the state for his first appearance on Friday, headlining a dinner for the state party that the chairman said had broken fund-raising records. More than 500 people attended, arriving from across New Hampshire and beyond, as Trump loyalists waved flags outside the downtown Manchester hotel.

“If I had a magic wand, I would like Trump,” said Sue Higgins, 53, a dental hygienist from Belknap County, a conservative stronghold in central New Hampshire. “He’s the only person who has the chutzpah to save America. But I’m not sure he’s the most electable.” As she waited for Mr. DeSantis to speak, she said she might support Mr. Trump again anyway.

Allison Chaffee, 36, who drove up two hours from Massachusetts to see Mr. DeSantis, described herself as an emissary “from the group that sways elections: suburban moms.” And her message was to move on.

“I hear what the moms say,” she said. “They are speaking Republican and then they vote Democrat. They only just hate Trump.”

But Lynda Payette, 68, of Bethlehem, N.H., waved away any talk of Mr. Trump’s vulnerabilities. “I believe that God placed him there and no man is going to take him down,” she said. “He’s electable, same as 2015.”

Then she turned the electability question on Mr. DeSantis, pointing to his decision to sign a six-week abortion ban, which she called extreme. “I really think we’ve got to give a little on this abortion thing,” she said, as a friend nodded in agreement.

For the sizable faction of the G.O.P. that has swallowed Mr. Trump’s falsehood that the 2020 election was stolen, electability is a particularly moot argument. They don’t think he lost.

If an anti-Trump electability message were to gain ground anywhere, it might be New Hampshire, where multiple competitive seats were fumbled away in 2022 by Republican candidates whom party leaders had warned were out of the mainstream, including in a marquee Senate race.

The state’s governor, Chris Sununu, who has teased a possible 2024 bid of his own, warned that Don Bolduc, the Republican who ran for Senate in New Hampshire, was a “conspiracy-theory type” candidate who would lose. Mr. Bolduc won the primary and lost in November.

“What I hear from some of our activists is we’re tired of losing,” said Christopher Ager, who took over the chairmanship of the New Hampshire Republican Party in a contested fight.

Fergus Cullen, a former New Hampshire Republican Party chairman, is less sanguine. He is the lone elected Republican on the City Council in Dover, where he needs to win over Democratic voters in a ward carried by Mr. Biden.

“I don’t see any signs of pragmatic, strategic voting among primary voters,” said Mr. Cullen, a critic of Mr. Trump. “I fully believe he has the ability to get the lemmings to follow him off the cliff again, no matter how far down it goes.”