Justin Penn, a Pittsburgh voter who calls himself politically independent, favored Joseph R. Biden Jr. in a matchup with President Trump until recently. But the president’s performance during the coronavirus outbreak has Mr. Penn reconsidering.

“I think he’s handled it pretty well,” he said of the president, whose daily White House appearances Mr. Penn catches on Facebook after returning from his job as a bank security guard. “I think he’s tried to keep people calm,” he said. “I know some people don’t think he’s taking it seriously, but I think he’s doing the best with the information he had.”

Although Mr. Penn, 40, said he did not vote for Mr. Trump, his opinion of the president has improved recently and he very well might back him for a second term.

Across the country, the coronavirus has sickened more than 150,000 people, cost millions their jobs and tanked the stock market. Yet the president’s approval ratings are as high as they have ever been, despite what most agree to be his slow performance dealing with the crisis, as well as his record of falsehoods about the virus, his propensity to push ideas and treatments that contradict expert advice, and his habit of lashing out at governors on the front lines.

While public perceptions are fluid in a crisis, a notable twist in polling at this point is that independents are driving Mr. Trump’s bump in approval, and some increased Democratic support is a factor as well. Gallup called that “highly unusual for Trump” in reporting its latest survey, which was released last week and showed Mr. Trump’s approval rating at 49 percent, equal to the best of his presidency.

While Republicans’ views of Mr. Trump were flat — a sign they had already topped out — approval by independents rose by eight percentage points from early March, while Democratic approval was up by six percentage points.

Polling experts said that it was normal for the country to rally around a president during a national crisis, and that Mr. Trump’s dominance of the airwaves alone was enough to sway a slice of voters who don’t normally tune in to politics.

“There are people who haven’t even heard Trump that much, while the rest of us have been obsessed,” said Matt Grossmann, director of the Institute for Public Policy and Social Research at Michigan State University. “Those people are paying attention and seeing Trump a lot.”

Every modern president has seen their approval surge after significant national crises, although those bumps have diminished in size in recent administrations, as the country’s politics became more polarized. President Barack Obama gained just seven points after U.S. forces killed Osama bin Laden in 2011. The rally-round-the-flag effect is also often fleeting. President Jimmy Carter’s approval nearly doubled in 1979 when Iran seized American hostages, but as the crisis dragged on for more than a year, Mr. Carter’s approval plummeted and he lost re-election.

Interviews with about two dozen independent and Democratic voters, most of whom said they “somewhat disapproved” of Mr. Trump in a poll last year by The New York Times and Siena College, showed that some now expressed more positive views of him. Their numbers were small, consistent with what pollsters say is by historical standards a modest bump in approval for a commander in chief during an emergency.

Kathleen Mathien, an independent in Maricopa County, Ariz., said that she did not vote for Mr. Trump, but that her opinion of him had risen during his White House appearances to talk about the virus.

“He’s not one to be bullied,” she said, adding that she also saw flashes of empathy, a trait many critics find lacking in Mr. Trump.

Ms. Mathien, 64, a designer of cabinetry, explained that she doesn’t closely follow politics and finds it difficult to get a true understanding of candidates beyond the “smoke and mirrors” they project. “It’s so hard sometimes to vote if you don’t know who the real person is,” she said. Undecided as of now, she said Mr. Trump has a chance to win her vote.

Last week, a Monmouth University poll showed the president’s overall approval at 46 percent, an improvement driven in part by higher Democratic support. Patrick Murray, the director of the university’s Polling Institute, called the shift by some Democrats “microscopic in polling terms.”

“Any other president and we would expect those job ratings to swing by more than 10 points because of the situation,” Mr. Murray said.

Mr. Trump’s ratings lag far behind many of the nation’s governors, who have seen a sharp increase in their approval ratings as they rush to contain the virus. Unlike Mr. Trump’s, their approval ratings do not show the same level of partisan divide.

More than seven in 10 voters in states with a significant number of coronavirus cases gave their governor a positive review in the Monmouth survey. Even in states with the fewest reported cases, 61 percent of Americans said their governor was doing a good job.

Still, small shifts in Mr. Trump’s approval could make a difference in a close-fought general election. A Washington Post/ABC poll this past weekend showed Mr. Trump improving on a seven-point deficit against Mr. Biden a month ago to reach a near tie with the former vice president, 49 percent to 47 percent.

“President Trump has broken through the narrow range of 42 to 46 percent approval where he’s been for the last two years and indeed for much of his presidency,” said Whit Ayres, a Republican pollster. “It’s an open question whether those people who are changing now would actually vote in a different way in November. Some of the independents may. I doubt that many of the Democrats will.”

Anna Greenberg, a Democratic pollster, said, “I would be a little careful in whether it translates into something permanent,” adding, “The challenge for Trump is that he’s inconsistent.”

Robert Taylor, 31, a computer programmer in York County, Pa., wants Senator Bernie Sanders to be the Democratic nominee and is unsure if he will vote for Mr. Biden in a contest against the president.

“I’m not one of those people who hate Trump and thinks everything he’s done is wrong,” he said. He could vote for Mr. Trump if the president successfully leads the country through the coronavirus crisis, Mr. Taylor said. “We’ll see how he handles everything from here on out.”

Two months ago, Neil Ferguson of Earling, Iowa, stood in a corner for Senator Amy Klobuchar at Iowa’s Democratic caucuses. But he is displeased today by Democrats criticizing Mr. Trump’s leadership and wants the country to rally around the president at a time of national emergency.

“At some point we’ve got to get behind this together,” he said. “Every step of the way he’s criticized,” he said of the president. “I know a lot of voters out here that say yeah, had they given the guy a chance, maybe things would have been a lot better.”

Mr. Ferguson, 67, who is retired from the military, regularly watches the White House briefings, and though he sometimes winces over the president’s rambling delivery, he is impressed with Vice President Mike Pence and with Mr. Trump’s responses to reporters.

“When he gets to the question-and-answer period, he is pretty point-on,” he said. Four years ago Mr. Ferguson voted for a third-party candidate, but this year he has decided to vote for the president.

Among voters already supportive of Mr. Trump, recent polling shows their enthusiasm to support him in November is running well ahead of the enthusiasm of Biden backers. Janice Friedel, a professor in Des Moines and a Democrat, liked Mr. Trump before the virus hit, and now her support has grown stronger.

“I thought President Trump was doing OK, but this really has brought out his strong leadership, his ability to bring people together across the aisle,” she said. “I am a Democrat, but I am going to vote for him. I don’t see leadership on the Democratic side. But I certainly will vote for Trump.”

There are some Democrats and independents who were initially inclined to give the president the benefit of the doubt over the coronavirus, but have since concluded that he is failing.

“In the beginning, when he went on TV he sounded very presidential, sounded like he wanted to get in front of this,” said Francis Newberg of Delaware County, Pa. “I told my wife, ‘Listen to this guy, he sounds real.’”

But Mr. Newberg’s opinion swiftly went downhill as he watched the president attack Democratic governors and say that “everything is fine.”

“It’s not fine,” said Mr. Newberg, who lives with his wife outside Philadelphia in a community for residents over 62. Its three restaurants have closed and staff members now deliver three days of groceries at a time to residents.

“We had our first case of coronavirus diagnosed in our community,” said Mr. Newberg, who retired from a phone company. “There’s 1,800 of us. If it breaks out in here, there’s going to be a lot of boxes outside.”

In Florida, Jason Berger, an independent voter, told the Times/Siena poll last year that he strongly approved of the president. But Mr. Berger, a pharmacy technician, had an about-face as he watched Mr. Trump’s handling of the outbreak.

“The biggest pivot point for me was when he mentioned the cruise liner which held Americans, which he didn’t want to dock in California because he didn’t want the numbers to go up,” Mr. Berger said, referring to the Grand Princess cruise ship, which was held offshore with 21 infected people aboard in early March. “I found that extremely insulting. Those were Americans and they were sick.”

Four years ago, Mr. Berger, 46, did not vote in the presidential election, deciding to leave the outcome to voters who were more tuned in than he was at the time. He now regrets that decision. “I take full responsibility,” he said. He does not intend to sit out another race and will vote for the Democratic nominee in November.

“We need the government to take care of us in a crisis situation,” he said.