WASHINGTON — Only about half of Democrats think President Biden should run again in 2024, a new poll shows, but a large majority say they’d be likely to support him if he became the nominee.

The poll by the Associated Press-NORC Center for Public Affairs Research shows that 26 percent of Americans overall want to see Mr. Biden run again — a slight recovery from the 22 percent who said that in January. Forty-seven percent of Democrats say they want him to run, also up slightly from only 37 percent who said that in January.

The ambivalence among Democratic voters comes as Mr. Biden is preparing to formally announce his 2024 re-election campaign as soon as next week, according to people briefed on the discussions. The president has been eyeing April 25 — four years to the day since he entered the 2020 race — though no final decisions have been made.

Despite the reluctance of many Democrats to see Mr. Biden run for another term, 78 percent of them say they approve of the job he’s doing as president. A total of 81 percent of Democrats say they would at least probably support Mr. Biden in a general election if he is the nominee — 41 percent say they definitely would and 40 percent say they probably would.

Interviews with poll respondents suggest that the gap reflects concerns about Mr. Biden’s age, as well as a clamoring from a younger generation of Democrats who say they want leadership that reflects their demographic and their values. Mr. Biden, now 80, would be 82 on Election Day 2024 and 86 years old at the end of a second presidential term. He is the oldest president in history.

Jenipher Lagana, 59, said she likes Mr. Biden, calling him an “interesting man” who has had an “incredible political career.” She praised Mr. Biden for providing a “breath of fresh air” and said she approves of how he’s been doing his job as president.

Yet “my problem with him running in 2024 is that he’s just so old,” Ms. Lagana, who is retired and lives in California, said. “I would love to see somebody younger, like [Pete] Buttigieg or [Gavin] Newsom be able to get in there and handle things maybe a little differently just because they’re a younger person.”

Donna Stewart, 48, a program director for a nonprofit in New York, also pointed to Mr. Biden’s age as a concern.

“I voted for him. I like him as a person. I like him as a leader for the country,” she said. “However, I just feel that he’s still lacking the up-to-date knowledge of what needs to be done.”

During the 2020 presidential campaign, Mr. Biden appeared to hint that he would limit himself to just one term in the White House, framing his candidacy as a bridge to a new generation of Democratic leaders. Yet while in office, Mr. Biden has made his intentions clearer that he would run again for a second term, saying as recently as last week in Ireland that he’s “already made that calculus” and that the announcement will happen “relatively soon.”

With only nominal primary challengers and a chaotic Republican field, the president and his senior aides have felt little pressure to formalize a re-election campaign. Instead, Mr. Biden has focused on governing, holding events at the White House and traveling across the country to sell his top legislative achievements such as a bipartisan infrastructure law and a sweeping climate, health care, and tax package.

The president and his senior political advisers are meeting with Democratic donors in Washington next week in an event meant to energize the party’s top contributors ahead of Mr. Biden’s expected re-election campaign.

Mr. Biden has also batted away questions about his age, saying that voters simply need to “watch me” to determine whether he’s up to the job as president.

While many Democrats remain tepid on Mr. Biden because of his age, others said it was actually an asset.

Stephen Foery, 47, said Mr. Biden’s decades in Washington — first in the Senate and then as vice president — proved to be valuable in the first two years of his presidency “because he’s done a lot to fix the country in a very short amount of time.”

“I think that one of the benefits of living a long life is that you have a lot of wisdom to impart,” Mr. Foery, a creative services manager in Pennsylvania, said. “If you gain as much experience as Biden has throughout his life, it would be a shame to simply disregard him because of his age.”

Mr. Biden’s job approval rating stands at 42 percent, a slight improvement from 38 percent in March. The March poll came after a pair of bank failures rattled an already shaky confidence in the nation’s financial systems, and Mr. Biden’s approval rating then was near the lowest point of his presidency. Thirty percent of Americans call the national economy good, a slight improvement from 25 percent a month ago.

Younger Democrats remain a reluctant part of Mr. Biden’s coalition — just 25 percent of those under age 45 say they would definitely support Mr. Biden in a general election, compared with 56 percent of older Democrats. Still, an additional 51 percent of younger Democrats say they would probably vote for Mr. Biden in a 2024 general election.

“It’s really hard to support somebody who is such a career politician, who has a vested interest in maintaining the status quo when the status quo doesn’t work for me,” Otis Phillips, 20, who lives in Washington state, said.

Mr. Phillips, a student, said he has been pleased with some of Mr. Biden’s initiatives, including his student loan forgiveness program and his focus on climate policy. Yet he emphasized: “I don’t like maintaining the status quo. And so I want things to change, and I don’t think Biden’s how we’re going to get that in the next four years.”

Both the current and former president could face resistance from the public as a whole in a general election. A total of 65 percent of U.S. adults say they would definitely or probably not support President Trump if he is nominated in a general election, including 53 percent who say they definitely would not. Mr. Biden’s obstacles are smaller by comparison but still substantial: 56 percent of Americans say they would be unlikely to support Mr. Biden in a general election, including 41 percent who say they would definitely not.

Mr. Biden has long bet that once voters are presented with a binary choice — either him or a Republican candidate, particularly if it is Mr. Trump — that a majority of the electorate will side with Democrats. He often quotes his father, Joseph R. Biden Sr., in his public remarks: “Joey, don’t compare me to the Almighty. Compare me to the alternative.”

“The only reason why I would not want him to run is because of his age. Like, that’s the only thing,” Shakeen Magee, 45, a self-employed Georgia resident, said.

Yet she said that if Mr. Biden does officially become the Democratic nominee in 2024, she would definitely support him “because we can’t take another Trump.” Ms. Magee added that “if we were to get another Republican in that office, it would just undo the little progress that Biden has been able to make.”

By Seung Min Kim and Emily Swanson. Zeke Miller and Hannah Fingerhut contributed.