CRESTON, Iowa — The voters at campaign events for Joseph R. Biden Jr. here in Iowa and across the country aren’t just shopping for a candidate for themselves.
As they jostle to take pictures with the former vice president and listen to him preach about national unity, they are often thinking about someone else — a dad, a neighbor or a colleague. They consider the political leanings of people close to them who are uncomfortable with the most liberal presidential contenders, but who hate the chaos of the Trump era and are receptive to the kind of centrist, seasoned candidacy Mr. Biden offers.
Some Democrats have been warning the party not to obsess over these potential swing voters, arguing that electability calculations about mythical undecided moderates are futile at this moment of extreme political polarization.
But for many Biden supporters, those voters are their Republican-leaning relatives and friends. And their perspectives are an increasingly prominent consideration as the Iowa caucuses near.
“I think he could get the independents and moderate Republicans who refuse to vote for Donald Trump,” said Bailey Smith, 27, a leader in Atlantic, Iowa’s business community and an undecided voter who attended a Biden campaign event on Sunday. Asked whether she had any moderate Republicans in mind, she replied, “My dad.”
It’s a dynamic that helps explain why, despite Mr. Biden’s series of missteps and uneven debate performances, many Democratic voters still believe the former vice president would stand the best chance against President Trump in a general election, polls show.
That’s a message Mr. Biden’s top surrogates are sounding at every turn, citing specific potential swing voters in the process.
“I always say that in Mount Pleasant, Iowa, if I didn’t have Republican friends, I wouldn’t have had any friends at all growing up,” said Christie Vilsack, the former first lady of Iowa and a prominent Democrat, as she introduced Mr. Biden at a campaign stop here on Saturday night. “Who will appeal to independents? And I want my candidate to be able to appeal to my Republican friends as well.”
For some Democrats, that development amounted to another reason for skepticism of Mr. Biden’s emphasis on bipartisanship, and his claims that Republicans might have an “epiphany” with Mr. Trump out of office.
Away from Capitol Hill, crossover voters are also a rare breed. But recent New York Times Upshot/Siena College surveys show that a narrow slice of swing voters does exist in key battleground states, as do voters who say they would be comfortable with Mr. Biden but not Senator Elizabeth Warren of Massachusetts, who is more progressive. Mr. Biden continues to lead the national polls, too, though he has struggled in the early voting states of Iowa and New Hampshire this fall.
“There are a few candidates — Biden, obviously, being one — that could be attractive to disaffected Republicans and more centrist Republicans,” said former Representative Carlos Curbelo, Republican of Florida. He also mentioned Mayor Pete Buttigieg of South Bend, Ind., former Mayor Michael R. Bloomberg of New York and Senator Amy Klobuchar of Minnesota.
Mr. Biden’s appeal to what some might call the “reasonable Republican dad” vote goes like this: He shuns far-reaching proposals like “Medicare for all”; he has a history of working with Republicans; his warm personal style is disarming; and he represents a return to what some moderates view as a more stable era.
A number of those factors are a liability among progressive voters, who are clamoring for bold change. But on the campaign trail with Mr. Biden, it’s the centrist Republican and independent voters who are often top of mind for his supporters as they think about the general election.
And according to a dozen Republican donors and strategists, as well as former elected officials who can speak free from concerns about retribution from Mr. Trump, there is some evidence that Mr. Biden can connect with these voters.
“Oh yeah, he definitely does,” said former Gov. John Kasich of Ohio, a 2016 Republican presidential candidate, pointing to Mr. Biden’s ability to speak to blue-collar voters. “He is a guy who can do it probably better than any other Democrat.”
Chuck Hagel, a Republican who served as defense secretary in the Obama administration, said he wrote in Mr. Biden’s name on the ballot instead of voting for Mr. Trump or Hillary Clinton in 2016. This year, he said, he favors Mr. Biden again, and would not back Mr. Trump — or Ms. Warren, or Senator Bernie Sanders of Vermont — in the general election. It’s a view shared by other Republicans, Mr. Hagel added.
“They’ve said to me, ‘If Biden is the nominee, I will vote for Biden, I will not vote for any of the other Democrats,’” recalled Mr. Hagel, a former Republican senator from Nebraska, citing Mr. Biden’s experience and his empathy. “I don’t know how big or deep or wide that is in this country, but I hear it.”
At Biden campaign events, prominent supporters often instruct attendees to think in pragmatic terms, a stark departure from the fiery populist messages that animate events held by Ms. Warren and Mr. Sanders.
“It’s not going to be you, it’s not going to be me, it’s not going to be the party faithful that turn this election — it’s going to be independents and moderate Republicans,” Chris Louscher, a Democratic activist, told a crowd at a Biden event in Algona, Iowa, this month, thinking of her brother-in-law in Florida. “I have a lot of family, a lot of friends, I have people in this audience today that are Republicans. They will vote for Joe Biden and that’s how he wins this next election.”
In recent days, Mr. Biden has been sharpening the contrast between himself and his Democratic rivals, using an addition to his stump speech to swipe at those who suggest that he is “naïve” for wanting to cooperate with Republicans — the very Republicans, he acknowledges, who are attacking him, his family and his “only living son.”
On Friday, a reporter asked Mr. Biden for evidence that Republican officials and voters had any interest in working with him.
“I think Republican voters have interest in finding common ground,” he said. “Wherever I go, there’s an enormous number of independents and Republicans who know and think we have to find common ground.”
It’s not just Republican votes that Mr. Biden is seeking. Several longtime Republicans have donated to his campaign and held events for Mr. Biden, too.
On Friday, Mr. Biden joined about two dozen people — roughly half of them Republicans — at a fund-raiser in Bel Air, Calif., hosted by Harry Sloan, a longtime Republican donor who supported Mrs. Clinton in 2016.
“I’ve spoken to many Republicans who don’t intend to vote for Trump,” Mr. Sloan said. “They’re looking for an alternative. They are pretty polarized against Warren and Sanders and that so-called progressive wing of the party.”
“And when those conversations come up, they tend to say, ‘But I would vote for Joe Biden,’” Mr. Sloan said, adding that his event brought in $100,000.
Yet there are clearly significant limitations to Mr. Biden’s appeal across the aisle — and to Democrats’ guesses about how Republicans might feel.
Some Democrats in the Des Moines area said they thought Douglas E. Gross, a prominent Iowa Republican, would be inclined to support a moderate like Mr. Biden.
That was a surprise to Mr. Gross.
“They’d have to do better than Joe Biden,” he said in an interview this month. “I just think his time has passed him by.”
Then there was Marie Hansen of De Soto, Iowa, who said she was “95 percent” decided on Mr. Biden. But at a Sunday night campaign event, she also described a friend — a now-disaffected Trump voter — who has been swayed by Mr. Trump’s criticisms of Mr. Biden and his son, Hunter. Ms. Hansen wondered about “just finding that candidate she can settle with.”
Many Democrats argue that it is not enough to settle for a candidate who could appeal to disaffected Trump voters or moderate Republicans. They say that without liberal enthusiasm, the party can’t compete against Mr. Trump and his intensely committed base.
Polls show several of the Democratic candidates doing well against Mr. Trump in hypothetical general election matchups. While the Biden campaign and his allies are making explicit overtures to Republicans, supporters of his leading opponents note that their messages, too, can cross strict party lines.
Representative Andy Levin, a Michigan Democrat who is supporting Ms. Warren, said that her “plan to end the corrupt system of self-dealing of the super rich is music to Republican, independent and Democratic ears.”
Mr. Sanders, who has gained in recent polls, appeals to younger progressive voters as well as some of the blue-collar constituents both Mr. Biden and Mr. Trump are seeking to engage.
“Iowans watched Donald Trump win by running as a fake Bernie Sanders, and they know the best way to beat the imitation is with the real thing,” said Bill Neidhardt, Iowa deputy state director for Mr. Sanders.
For all of the talk at Biden events about engaging independents and Republicans, Democratic attendees who hail from conservative areas tend to be the most skeptical that any Republicans are listening.
Dianne Ballard, 63, of Centerville, Iowa, said she thought that Mr. Biden could defeat Mr. Trump, but also noted that many Republicans negatively associate him with former President Barack Obama. Ms. Ballard, a Biden supporter, did not plan to press the case with her husband, a Republican, over Christmas.
“This is kind of Trump country, unfortunately,” Ms. Ballard said. “You’re either one or the other.”