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But there was quiet concern within the campaign, where aides took note of daily tracking numbers from Rasmussen Reports, a typically rosy assessment of how the president is faring, that showed support falling after the tax story. Among Mr. Trump’s circle, there was finger-pointing about how the issue was handled and a hesitancy to discuss with him an issue they know he is sensitive about.

Many of the president’s advisers argued that such stories have never harmed his standing with core supporters in the past and that this would be no exception. They recognized, however, that Mr. Trump would have to have an effective response prepared for the debate, the first encounter between the two candidates, scheduled for 9 p.m. Eastern at Case Western Reserve University in Cleveland. Some aides suggested he try to finesse an answer similar to what he has said previously, that taking advantage of the tax code is simply “smart” for a business person, while also painting himself as a jobs creator.

Tim Murtaugh, the campaign spokesman, called the report inaccurate because the president has paid “tens of millions in taxes,” without specifying whether they were federal income taxes or directly denying that there were years he paid none. “This has been litigated in front of the voters before,” he said on Fox News. “The president released more than 100 pages of financial records. And Americans made their judgment in 2016 and elected him president. There’s nothing in there that changes anyone’s mind.”

Still, some of the details, particularly paying only $750 in federal income taxes two years in a row and deducting hairstyling expenses, among other things, could resonate in a visceral way. Democrats expressed hope that they could fuel a sense of injustice with everyday Americans who pay far more themselves and cannot write off hair expenses.

Representative Alexandria Ocasio-Cortez of New York tweeted that she paid thousands of dollars in federal taxes in 2016 and 2017 — when she was still working as a bartender in New York City. “He contributed less to funding our communities than waitresses & undocumented immigrants,” she wrote. The Biden campaign video showing the typical income tax paid by various workers amplified the attack.

The Biden campaign also began selling T-shirts, buttons and stickers that say, “I paid more income taxes than Donald Trump.” And the campaign on Monday launched an online “Trump tax calculator” allowing people to calculate how much more they paid in federal income taxes than Mr. Trump. “Mad? Us too,” the web page for the tax calculator said. “Join our campaign to elect Joe Biden and make ‘billionaires’ like Donald Trump pay their fair share.”