His verbal miscues are nothing new, friends note; he has struggled throughout his life with a stutter and was a “gaffe machine,” to use his own term, long before he entered Social Security years. Advisers said his judgment is as good as ever. So many of them use the phrase “sharp as a tack” to describe him that it has become something of a mantra.

Mr. Biden says age is a legitimate issue but maintains that his longevity is an asset, not a liability. “You say I’m ancient?” he said at the White House Correspondents’ Association dinner in April. “I say I’m wise.”

Still, few people fail to notice the changes in one of the nation’s most public people. As vice president a dozen years ago, Mr. Biden engaged in energetic squirt gun battles each summer with the children of aides and reporters. More than a decade later, he shuffled stiffly across the Edmund Pettis Bridge in Selma, Ala., to mark the anniversary of Bloody Sunday.

Polls indicate the president’s age is a top concern of Americans, including Democrats. During a recent New York Times focus group, several voters who supported Mr. Biden in 2020 expressed worry, with one saying: “I’ve just seen the blank stare at times, when he’s either giving a speech or addressing a crowd. It seems like he loses his train of thought.”

Unease about Mr. Biden’s age suffuses Democratic circles. One prominent Wall Street Democrat, who like others spoke on the condition of anonymity to avoid offending the White House, noted that among party donors it was all anyone was talking about. At a small dinner earlier this year of former Democratic senators and governors, all of them in Mr. Biden’s generation, everyone at the table agreed he was too old to run again. Local leaders often call the White House to inquire about his health.

In private, officials acknowledge that they make what they consider reasonable accommodations not to physically tax an aging president. His staff schedules most of his public appearances between noon and 4 p.m. and leave him alone on weekends as much as possible.

A study of Mr. Biden’s schedule based on data compiled by Axios and expanded by The New York Times found that Mr. Biden has a similar morning cadence as the president he served, Barack Obama. Neither had many public events before 10 a.m., just 4 percent in Mr. Obama’s last year in office and 5 percent in Mr. Biden’s first two and a half years. But the real difference came in the evening. Mr. Obama was twice as likely to do public events after 6 p.m. compared with Mr. Biden, 17 percent to 9 percent.