A new poll out Sunday suggests that more than two-thirds of Americans are concerned that President Biden lacks the necessary mental and physical health to be president — and more than half of them describe those concerns as “major.”

The poll, by NBC News, found that 68 percent of Americans are concerned about Mr. Biden’s acuity, and 55 percent of them say they have “major concerns” about it. That’s compared to 55 percent of Americans who say they have similar concerns about President Trump, including 44 percent who have “major concerns” about him. Three-fourths of respondents to the poll said America is on the wrong track, and only about 43 percent expressed approval of the job Mr. Biden is doing as president.

Amid the dismal numbers for the incumbent, the poll suggests that his likely challenger, Mr. Trump, has widened his lead among the Republican candidates since he was indicted earlier this month on charges of illegally hoarding classified documents at his estates in Florida and New Jersey. Six out of 10 Republican respondents to the poll said the charges and investigations against Mr. Trump are politically motivated. Mr. Trump’s 29-point lead against Governor DeSantis is up five points compared to a similar April survey.

“For the first time in history, a former president has been indicted, and we can’t find a marker in this survey that it’s had an impact with his standing,” a Republican pollster with Public Opinion Strategies, Bill McInturff, who conducted the survey with a Democratic pollster, Jeff Horwitt of Hart Research Associates, told NBC. “Not only are they sticking with Trump post-federal indictment,” Mr. Horwitt said of Republican voters, “there are several signs that his support is growing or others are losing ground, particularly Ron DeSantis.”

The NBC News poll was conducted June 16-20, a week after the federal grand jury returned with charges against Mr. Trump, and surveyed 1,000 registered voters across the country.

The problem for Mr. Trump is not among Republican voters, who clearly still favor him over most of the ever-increasing Republican field, but among voters in the general election. In the poll’s first hypothetical test between Messrs. Trump and Biden, Mr. Biden leads by four points. Swap out Mr. DeSantis for Mr. Trump in that hypothetical, and the two candidates are tied.

“Looking back at 2020, the election was a referendum on Donald Trump,” the Democratic pollster, Mr. Horwitt, said. “And if we have a Biden-Trump rematch, there are powerful signs that the focus will once again be more on Trump than Biden.”

Advisers to the president reportedly have told him that he should embrace his age with a side of humor on the campaign trail, according to a report in Sunday’s Wall Street Journal. Movie mogul Jeffrey Katzenberg, an informal advisor to Mr. Biden’s campaign, says that if 80-year-old Harrison Ford can star in a new Indiana Jones movie and the 80-year-old front man of the Rolling Stones, Mick Jagger, can prance around a stage, then Biden should lean into his longevity as a sign of wisdom and experience. Mr. Biden would be 82 on inauguration day in 2025 if he wins re-election.

At a campaign event Saturday night at Washington, D.C., Mr. Trump embraced the accusations against him, calling them a campaign by “radical left Democrats, Marxists, Communists and fascists” that he considers a “great badge of courage.”

“For seven years, you and I have been fighting side-by-side to rescue our country from evil and from the sinister forces who hate it,” Mr. Trump told the crowd at a Faith and Freedom Coalition event. “Together, we’re warriors in a righteous crusade to stop the arsonists, the atheists, globalists and the Marxists — and that’s what they are — and we will restore our Republic as one nation under God with liberty and justice for all.”