Party leaders have rallied behind the president’s re-election bid, but as one top Democratic strategist put it, “The voters don’t want this, and that’s in poll after poll after poll.”As President Biden shifts his re-election campaign into higher gear, the strength of his candidacy is being tested by a striking divide between Democratic leaders, who are overwhelmingly unified behind his bid, and rank-and-file voters in the party who harbor persistent doubts about whether he is their best option.From the highest levels of the party on down, Democratic politicians and party officials have long dismissed the idea that Mr. Biden should have any credible primary challenger. Yet despite their efforts — and the president’s lack of a serious opponent within his party — they have been unable to dispel Democratic concerns about him that center largely on his age and vitality.The discord between the party’s elite and its voters leaves Democrats confronting a level of disunity over a president running for re-election not seen for decades.Interviews with more than a dozen strategists, elected officials and voters this past week, conversations with Democrats since Mr. Biden’s campaign began in April, and months of public polling data show that this disconnect has emerged as a defining obstacle for his candidacy, worrying Democrats from liberal enclaves to swing states to the halls of power in Washington.Mr. Biden’s campaign and his allies argue that much of the intraparty dissent will fade away next year, once the election becomes a clear choice between the president and former President Donald J. Trump, the dominant leader in the Republican primary field.But their assurances have not tamped down worries about Mr. Biden from some top Democratic strategists and many of the party’s voters, who approve of his performance but worry that Mr. Biden, who will be 82 on Inauguration Day, may simply not be up for another four years — or even the exhausting slog of another election.“The voters don’t want this, and that’s in poll after poll after poll,” said James Carville, a longtime party strategist, who worries that a lack of enthusiasm for Mr. Biden could lead to lower Democratic turnout in 2024. “You can’t look at what you look at and not feel some apprehension here.”James Carville, a longtime Democratic strategist, is among those who worry that the party’s voters are not enthusiastic enough about Mr. Biden’s re-election campaign.Raul E. Diego/Anadolu Agency, via Getty ImagesIn recent days, a barrage of grim news for Mr. Biden, including an autoworkers strike in the Midwest that poses a challenge to his economic agenda and the beginning of impeachment proceedings on Capitol Hill, has made this intraparty tension increasingly difficult to ignore. Those developments come amid a darkening polling picture, as recent surveys found that majorities of Democrats do not want him to run again, are open to an alternative in the primary and dread the idea of a Biden-Trump rematch.A CNN poll released this month found that 67 percent of Democrats would prefer Mr. Biden not be renominated, a higher percentage than in polling conducted by The New York Times and Siena College over the summer that found half would prefer someone else.In quiet conversations and off-the-record gatherings, Democratic officials frequently acknowledge their worries about Mr. Biden’s age and sagging approval ratings. But publicly, they project total confidence about his ability to lead and win.“It’s definitely got a paradoxical element to it,” said Gov. Phil Murphy of New Jersey, a Democrat who is among a group of governors who put aside their national ambitions to support Mr. Biden’s re-election bid. “This is only a matter of time until the broad party, and broadly speaking, Americans, converge with the opinions of folks like myself.”Many party officials say that Mr. Biden is making a high-stakes bet that the power of incumbency, a good political environment for his party and the fact that Democrats generally like the president will eventually outweigh the blaring signs of concern from loyal supporters. Any discussion of an alternative is little more than a fantasy, they say, since challenging Mr. Biden would not only appear disloyal but would also most likely fail — and potentially weaken the president’s general-election standing.One Democratic voter who backed Mr. Biden in 2020, James Collier, an accountant in Houston, sees the situation slightly differently. He said he would like Mr. Biden to clear the way for a new generation that could energize the party’s base.“I think he’s a little — not a little — he’s a lot old,” Mr. Collier, 57, said. “I’m hoping he would in his own mind think, ‘I need to sit this out and let someone else do this.’”There are no indications that anyone prominent will mount a late challenge to Mr. Biden, though strategists working for other elected officials say that a number of well-known politicians would probably jump into the race if, anytime before the end of the year, the president signaled he was not running.The situation is almost the opposite of the Republican field, where Mr. Trump holds a commanding lead among the party’s base but remains far less beloved by a political class that fears his unpopularity among moderate and swing voters will lead to defeat in 2024.William Owen, a Democratic National Committee member from Tennessee, was full of praise for Mr. Biden and said he was puzzled by surveys that consistently showed the president struggling to win over Democratic voters.“I’m looking at all the polling, and I’m amazed that it has so little to do with reality,” he said in an interview this past week. “A big part of it is just pure ageism. The American people are prejudiced against old people.”Yet in describing his interactions with Democrats around Knoxville, which he represented for years in the Tennessee legislature, Mr. Owen said he could not escape questions about Mr. Biden’s health.“People ask me: ‘How’s Joe doing? Will he last another four years?’” Mr. Owen said. “That’s the real question. Will Joe Biden last another four years? I’m happy to say, yes, he will. He’s going to live to be 103.”Officials in Mr. Biden’s campaign insist that hand-wringing about his age is driven by news coverage, not by voters’ concerns. They dismiss his low approval ratings and middling polling numbers as typical of an incumbent president more than a year away from Election Day.A campaign spokesman cited articles about Democrats’ fretting about President Barack Obama before his second term and noted the limitations of polls so far from an election, suggesting that Mr. Biden had ample time to make his case.“President Biden is delivering results, his agenda is popular with the American people and we are mobilizing our winning coalition of voters well ahead of next year’s general election,” said Kevin Munoz, the spokesman. “Next year’s election will be a stark choice between President Biden and the extreme, unpopular MAGA agenda.”Lt. Gov. Austin Davis of Pennsylvania, who is Black and has issued public warnings about Mr. Biden’s standing with Black voters, said that simply casting the election as a referendum on Mr. Trump and his right-wing movement — as Mr. Biden’s campaign did in 2020 — would not be enough to energize the Democratic base. Mr. Davis has urged the White House to be more aggressive about highlighting the impact of Mr. Biden’s accomplishments, particularly with Black voters.“Everyone is kind of exhausted by the fight between Biden and Trump,” he said. “People really want to hear leaders talk about how they’re going to improve the lives of their families.”Lt. Gov. Austin Davis of Pennsylvania has cautioned Democrats against framing the 2024 election as a referendum on former President Donald J. Trump.Matt Rourke/Associated PressOther Democrats argue that Mr. Biden’s campaign must make clearer that the stakes are bigger than just the president.“It’s about showing people that the future of American democracy is at stake,” said Representative Jennifer McClellan of Virginia, who is a member of the Biden campaign’s national advisory board. “It’s not just about which president can get through the day without tripping or stumbling over their words, which everybody is going to do, but which president is going to lead this country forward in a way that helps people solve problems and keeps American democracy intact.”Faiz Shakir, the campaign manager for Senator Bernie Sanders’s 2020 presidential bid, said Mr. Biden needed to show voters that he was fighting for the American public, pointing to battles like his administration’s legal fight with pharmaceutical companies over their new Medicare pricing plan.“The question that I would want to answer is, is he is a strong leader?” Mr. Shakir said. “When people see he is a strong leader, they will feel different about his age. They will feel different about the economy. They will feel different about a lot of things.”Malcolm Peterson, 34, a waiter in St. Paul, Minn., said he thought Mr. Biden had done a good job tackling issues related to climate change during his first term, but worried about the president’s outlook for a second term.Jenn Ackerman for The New York TimesMalcolm Peterson, a waiter from St. Paul, Minn., whose foremost political concern is climate change, said he generally approved of Mr. Biden’s work as president and thought he had done a good job tackling environmental issues. But he said he worried about whether the president would be able to continue that work in a second term.“I just wonder, because he’s quite old, what does he look like in another four years?” Mr. Peterson, 34, said. “I’m not a doctor. I just know what I’ve seen.” More