Mr. Biden sought to dismiss any concerns at a news conference three weeks ago, telling reporters that he was up to the rigors of the campaign trail and the presidency when asked if he had taken a cognitive test. “I’ve been tested, and I’m constantly tested,” he said. “Look, all you’ve got to do is watch me, and I can hardly wait to compare my cognitive capability to the cognitive capability of the man I’m running against.”

Mr. Trump recently suggested injecting disinfectant to counter the coronavirus and he has been reported to once ask aides about using nuclear weapons to disrupt hurricanes.

Mr. Biden has the edge upstairs in the view of the public. In a poll by The Hill newspaper with the HarrisX research firm released this week, 56 percent of voters said the former vice president was mentally fit to lead the nation, while just 45 percent said that of Mr. Trump.

Other American presidents have struggled with cognitive or psychological issues but rarely addressed them publicly the way Mr. Trump has. A Duke University study in 2006 found that nearly half of presidents suffered from mental illness at some point in their lives. Nearly 25 percent of presidents met the criteria for depression, according to the study, including James Madison, Abraham Lincoln and Calvin Coolidge. Ronald Reagan’s aides were concerned enough that he was slipping late in his presidency that they secretly considered whether to invoke the 25th Amendment to remove him for incapacity before deciding he was still fit enough to finish his tenure.

In unscripted moments before the camera, the president sometimes wanders from topic to topic without completing a train of thought, repeats himself or falls back on the same familiar phrases when questioned. He makes statements that defy common sense, like insisting that if there were not so many coronavirus tests conducted, there would not be so many cases of the virus, which as skeptics have pointed out is roughly equivalent to saying that there would not be as many pregnancies if there were not as many pregnancy tests administered.