Democratic insiders in Washington and key primary states expect President Biden to follow through on his intention to stand for re-election and appear to have little appetite for casting him aside, though they expressed concerns about his advanced age and persistently low poll numbers.

The White House has repeatedly said that Mr. Biden, 79, the oldest president to be sworn into office, intends to run for re-election. A person familiar with the president’s advisers’ thinking said they are planning on him running and that the private conversations have matched the public statements. The current discussion is that an announcement would happen after the midterm elections, likely sometime in the spring of 2023, this person said.

That means it will likely be several months before the decision is official, during which Democrats could shift their views on their 2024 preferences. An announcement of another White House bid by former President Donald Trump could also affect the calculus of Mr. Biden and other Democrats.

Plummeting approval ratings, a stalled legislative agenda and surging inflation have fueled questions about whether Democrats would be better off in 2024 with a different standard-bearer. Roughly half of all adults and one-third of Democrats in a March Wall Street Journal poll said they didn’t think Mr. Biden would run again, with many citing his age as the main argument against him running. A Yahoo News/YouGov poll last week found 40% of 2020 Biden voters said he shouldn’t run again compared with 37% who said he should, a reversal from a month earlier.

Against that backdrop, some potential candidates are visiting key presidential-voting states and building a national profile in case Mr. Biden forgoes a re-election bid. But conversations with Democrats in Washington and in primary states found most don’t see a better or realistic option than the incumbent to top the 2024 ticket.

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Key Democrats also said they doubted many top-tier candidates would try to push Vice President Kamala Harris aside if Mr. Biden chose not to run, and early polls suggest she would likely be the initial front-runner in such an event. But her own poll numbers aren’t better than Mr. Biden’s, and Democrats have mixed views of how strong she would be leading a ticket. And some noted the party’s nomination fights in 2016 and 2020 included strong runs by liberal Sen. Bernie Sanders (I., Vt.), who has ruled out running against Mr. Biden but not Ms. Harris.

Speculation among Democrats has touched on other 2020 contenders—Minnesota Sen. Amy Klobuchar and current Transportation Secretary Pete Buttigieg —as well as Biden infrastructure czar Mitch Landrieu and Govs. Phil Murphy of New Jersey and Gavin Newsom of California, who has been taking a more assertive national role in sparring with Republicans on issues like abortion.

But there is no obvious person whom party leaders would see as the best pick if neither Mr. Biden nor Ms. Harris topped the ticket.

Illinois Gov. J.B. Pritzker made a weekend trip to New Hampshire, the state that typically hosts the second nominating contest in the presidential primary, after Iowa’s caucuses. He gave the keynote address at the state Democratic Party’s annual convention Saturday and on Friday attended its splashy annual fundraiser.

“I don’t know if this is 2024, 2028, 2032…but listen, we’re grateful to have you in New Hampshire,” Rep. Annie Kuster (D., N.H.) said Friday to Mr. Pritzker, eliciting laughter from a packed Manchester ballroom.

Illinois Gov. J.B. Pritzker speaking at New Hampshire Democratic Party's annual convention in Manchester, N.H., on Saturday.

Photo: Joshua Jamerson/The Wall Street Journal

These kinds of party events have been stops for past presidential hopefuls. Asked if he was considering a 2024 run, Mr. Pritzker, who is up for re-election this year, told reporters in Illinois: “I can’t tell you anything other than I love the job that I have.” He met with Democratic National Committee officials last week in Washington.

The governor intends to serve his full second term if he wins in November, according to a campaign official, adding that the trip was focused on electing Democratic governors. But Mr. Pritzker has stopped short of ruling out a presidential bid if Mr. Biden steps aside.

The significance of an out-of-state politician coming to New Hampshire wasn’t lost on the state’s establishment class, who took Mr. Pritzker’s attendance as dipping his toe in the water. “They don’t have another good explanation [to be] in New Hampshire. That’s it,” said Bob Backus, a former state lawmaker who attended the Friday fundraiser. He said he didn’t know much about the governor, but worried Democrats might undercut Mr. Biden if 2024 speculation kept up.

“I’m just worried that some of the Democrats are going to undercut Biden, and that’s what’s worrying me more than anything, because there’s already all these rumors,” Mr. Backus said. “I hope we don’t have it here in New Hampshire.”

Harry Judd, a former official in Bow, N.H., has hosted presidential candidates like John Kerry and John Edwards at his home for large house parties—a staple of the retail politics that has come to define New Hampshire’s primary. He doesn’t see those kinds of events happening soon but said small meet-and-greets—like an intimate dinner he attended with New Jersey Sen.

Cory Booker during his failed 2020 bid—might occur from time to time.

“As long as he’s saying he’s still running, others will lay back,” he said of Mr. Biden, predicting the field would largely stay that way “unless he steps aside.”

Still, Mr. Judd and his wife, New Hampshire state lawmaker Mary Beth Walz, both said they had reservations about Mr. Biden’s age.

“I am actually reasonably comfortable with how he’s handled the office,” Ms. Walz said, adding she would support him if he ran in 2024. But, she added, “I worry about what age will do to him. Not where he is now—I wonder where he’s going to be in six years.”

Cinde Warmington, the sole Democrat on New Hampshire’s Executive Council advising the governor, didn’t mention Mr. Biden by name at the Friday fundraiser but said Democrats should stick together and not fight each other—even in the face of expected electoral losses this fall. “No time for a gnashing of teeth. No time for anguish,” Ms. Warmington told Democrats.

A week earlier, Ms. Harris spoke at the South Carolina Democratic Party’s Blue Palmetto Dinner—her first appearance at a state party event as vice president. Ms. Harris previewed her midterm message contrasting Democrats and Republicans and reminisced about her time in the state while running for president in 2019.

“We see how South Carolina brings critical representation to the presidential nominating process,” Ms. Harris said of the state where the Democratic electorate is largely Black. “We see how South Carolina Democrats set President Joe Biden and me on a path to the White House.”

Rep. Jim Clyburn (D., S.C.), who paved the way for Mr. Biden to secure the Democratic nomination in 2020, suggested he would back her if Mr. Biden isn’t on the 2024 ticket.

“Right now, I’m for Biden, and second I’m for Harris—that’s one and two on the ticket, but that’s one and two in my heart as well,” he said. “So I don’t care who goes to New Hampshire or Iowa, I’m for Biden and then I’m for Harris—either together or in that order.”

Mr. Clyburn, who is 81 and asked often by reporters about his own retirement plans, said Mr. Biden should ignore concerns about his age and re-election. “My advice: be yourself, stay focused. Make the promises and keep them.”

Vice President Kamala Harris and Rep. James Clyburn (D., S.C.) at the South Carolina Democratic Party’s Blue Palmetto Dinner earlier this month.

Photo: Logan Cyrus/Agence France-Presse/Getty Images

Privately, other congressional Democrats have been less sanguine about Mr. Biden’s age and his overall performance in office. They have expressed frustration that he hasn’t acted more decisively in tackling inflation and pushing through legislative priorities before midterm elections that could take away the party’s majorities. Administration officials have pointed to several steps Mr. Biden has taken to try to curb record inflation, while acknowledging there was only so much they could do in the near term to lower prices.

Mr. Biden has also drawn negative attention in recent months for making spontaneous comments on important policy issues that his aides then had to walk back, as was the case after he said in March that Russian President Vladimir Putin “cannot remain in power.”

Mr. Biden has been headlining fundraisers for the DNC, which has raised $213 million for the midterms. It has invested heavily in battlegrounds like Arizona, Georgia, Michigan, Nevada, North Carolina, Pennsylvania and Wisconsin.

Democratic officials in South Carolina and New Hampshire said if Mr. Biden decides not to run, Ms. Harris, the first woman and Black vice president, would enter the Democratic nomination process with a significant advantage—especially if Mr. Biden endorsed her.

“I think they better think long and hard about trying to challenge Vice President Harris if she runs for president,” said Deyaska Spencer, chairwoman of the Democratic Party in Richland County, which includes Columbia, S.C. Ms. Spencer added that she still expects Mr. Biden to run.

Ms. Harris’s approval ratings, like Mr. Biden’s, have been sinking, but her supporters frequently point to her strong support among Black voters, who are key to winning the nomination. A recent Beacon Research/Shaw & Co. Research poll released last month by Fox News found 41% approved of her job performance, while 53% disapproved. But among Black voters, 73% approved of her.

She is also leading by a wide margin in several polls conducted this year of the Democratic primary in which Mr. Biden isn’t included. Still, some Democrats said they would be willing to listen to a strong challenger, should one emerge.

Sen. Bernie Sanders (I., Vt.) has ruled out running against President Biden but not against Vice President Kamala Harris.

Photo: Manuel Balce Ceneta/Associated Press

Pamella Coffey, 57, a Democratic voter and circuit court clerk from Chicago, said that as a Black woman she would like to be supportive of Ms. Harris running for president, but remains “unsure” about whether she would be the best nominee and would have to look at any future field of candidates.

Ms. Harris terminated her political operation when she took office. She has been wary of engaging in political activities that would hint at 2024 ambitions, according to people familiar with her thinking, but she is expected to ramp up her travel, campaigning and fundraising on behalf of Democrats and touting the administration’s accomplishments.

Mr. Sanders, who has mounted two previous presidential bids and is a leader among progressives, also hasn’t ruled out another presidential bid in the event of an open primary, according to a memo circulated by his former campaign manager in April. The memo noted Mr. Sanders’ favorable poll numbers and the power of his endorsement.

Mr. Sanders, 80, said this month, in an interview with CNN, that he wouldn’t run against Mr. Biden. He declined to answer when The Wall Street Journal asked Wednesday if he’d run against Ms. Harris. “This is 2022, so we’re worried about this election and I’m gonna do everything I can to make sure Democrats retain control of the House and the Senate,” he said.

In addition to Illinois, Mr. Sanders held events with union workers this week in Wisconsin and Iowa, which is lobbying to hold on to its first-in-the-nation presidential contest.

Andrew Thompson, 32, a Democratic voter and Black professor of public administration who lives in Chicago, attended a pro-union rally there with Mr. Sanders. Mr. Thompson said he would support Mr. Biden again, but begrudgingly. He’s unsure if he would support Ms. Harris for the nomination in lieu of Mr. Biden. “I haven’t seen much of her,” he said. “I expected more out of both of them.”

—Eliza Collins, John McCormick and Catherine Lucey contributed to this article.

Write to Joshua Jamerson at joshua.jamerson+1@wsj.com and Tarini Parti at Tarini.Parti@wsj.com