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WATERLOO, Iowa — Facing new pressure from Senator Bernie Sanders in Iowa, Hillary Clinton confronted him in direct terms on Monday, warning Iowans that if elected president, he would put their health insurance in the hands of Republican governors and raise taxes on the middle class.

Sharpening her tone as new polling suggested that Mr. Sanders was closing in on her in the state, Mrs. Clinton assailed Mr. Sanders by name, telling 300 people on a frigid and snowy day here, “I think it’s time for us to have the kind of spirited debate that you deserve for us to have.”

“We’re so much better than the Republicans,” she said, “but we do have differences and you deserve to know what those differences are.”

Mrs. Clinton also looked to repel criticism from Mr. Sanders that she is a patron of the rich, unveiling a “fair share surcharge” that would impose a new tax on people who earn more than $5 million a year. Her campaign estimated that raising the income tax on the wealthy by 4 percentage points would generate $150 billion in revenues over 10 years.

On health care, Mrs. Clinton said that she wanted to build upon the Affordable Care Act, while Mr. Sanders wanted to scrap it in favor of a universal health plan that would empower the states — and potentially Republican governors, who she has said cannot be trusted. This, along with the other progressive policies of Mr. Sanders, who represents Vermont, would require raising taxes on more than just the rich, she argued.

“There’s no way, if you do the arithmetic, to pay for what he has proposed without raising taxes on the middle class,” she said. “That’s where he and I part ways.”

Her remarks came a day after a new poll showed Mrs. Clinton’s lead in Iowa narrowing to three percentage points over Mr. Sanders. Mrs. Clinton’s campaign is counting on a victory here in the caucuses on Feb. 1 because the next contest is eight days later in New Hampshire, where Mr. Sanders has held a durable lead in polls.

“I feel really good about where we are, but I’m not taking anything for granted,” Mrs. Clinton said.

Her comments on Monday echoed the language she used in December 2007 when she told reporters in Cedar Rapids that “now the fun part starts” — a promise to make clear to Iowans the deficiencies of Barack Obama’s policies. She placed third in the Iowa caucuses, which Mr. Obama won.

“We’re into the last month, and we’re going to start drawing the contrasts, because I want every Iowan to have accurate information when they make their decisions,” she said at the time.

Mr. Sanders also campaigned in Iowa on Monday, similarly pressing an economic argument in trying to coax voters out to support him on caucus day. Speaking to reporters in Pleasantville, he said that Mrs. Clinton was attacking him because of polls that showed he was gaining support — and that she was realizing she could lose.

“Secretary Clinton and her campaign now know that she is in serious trouble,” Mr. Sanders said. “And, I think a candidate who was originally thought to be the anointed candidate, to be the inevitable candidate, is now locked in a very difficult race here in Iowa and in New Hampshire. Obviously, in that scenario, what people do is start attacking.”

Mr. Sanders highlighted how he has called for increasing Social Security benefits, and would pay for it by raising the cap on income that is eligible for taxation to finance the program. That way, the wealthy would pay more into the system.

“In 2008, as I recall, President Obama introduced very similar-type legislation,” Mr. Sanders said of his Social Security bill. “That legislation was then opposed by Senator Clinton, who was running against him. I would hope that now in the year 2016, when so many of our seniors and disabled vets are struggling, that Secretary Clinton will join me in demanding that we raise the cap on taxable income, that we expand benefits for our seniors and our disabled vets, and that we extend the life of Social Security for 50 years.”

As the candidates’s squabbled in Iowa, Vice President Joseph R. Biden Jr. made it clear he is paying close attention to the race. In an interview with CNN on Monday, Mr. Biden praised Mr. Sanders’s approach to income inequality, saying that he “is speaking to a yearning that is deep and real. And he has credibility on it.”

The issue, he added, was “relatively new for Hillary,” but added that she had “come forward with some really thoughtful approaches.”

Mrs. Clinton’s pointed remarks — and the sparring between the candidates — suggested that the Democratic presidential debate on Sunday could take a sharper tone than the previous ones, which were largely cordial affairs.

Fact Check: Hillary Clinton Says Law Bars All Suits Against Gun Industry Mrs. Clinton said legislation that Bernie Sanders supported in 2005 forbids all lawsuits against gun makers and sellers. Not all, but nearly. See all Election 2016 fact checks » After President Obama said last week that he would not support any Democrat who does not favor new gun control measures, Mrs. Clinton has tried to call attention to Mr. Sanders’s vote in 2005 for a bill that shields gun manufacturers and dealers from liability lawsuits. Mr. Sanders was a House member at the time; Mrs. Clinton, then a senator from New York, voted against the bill. On MSNBC’s “Hardball” on Friday, Mrs. Clinton claimed that the legislation said “no one can sue a gun maker or a gun seller,” and on CBS’s “Face the Nation” on Sunday, she said it provided the gun industry with “absolute immunity from any kind of responsibility or liability.” The bill, the Protection of Lawful Commerce in Arms Act, followed a flurry of lawsuits against the industry by municipalities. It shields gun manufacturers, sellers and their trade associations from liability lawsuits “resulting from the criminal or unlawful misuse” of firearms and ammunition. When President George W. Bush signed the measure into law, the National Rifle Association called it “the most significant piece of pro-gun legislation in 20 years.” But Mrs. Clinton is overstating the breadth of the law. "This statute does not grant blanket immunity to the firearms industry," said Timothy D. Lytton, a law professor at Georgia State University who edited the book "Suing the Gun Industry: A Battle at the Crossroads of Gun Control and Mass Torts." Though its protections are quite expansive, the law specifies six exceptions to the prohibition on civil actions. For example, a victim can sue a dealer who has been convicted of selling a firearm knowing that it would be used to commit a violent crime. Suits are also allowed in matters involving injuries or property damage resulting from manufacturing or design defects, as long as the firearm in question was not used in a criminal act. Another exception allows lawsuits when a gun maker or seller “knowingly violated a state or federal statute applicable to the sale or marketing of the product.” These exceptions are narrow, and they do provide broad protections to the gun industry, as Mrs. Clinton suggests. Whether the exceptions would ultimately prove meaningful was a subject of debate when Congress passed the legislation in 2005. “The aim of this bill is clear: to allow legitimate lawsuits against a manufacturer when the legal principles to do so are present,” Senator John Thune, Republican of South Dakota, said at the time. Senator Jack Reed, Democrat of Rhode Island, took the opposite view: “These exceptions in the bill have been carefully crafted to prevent lawsuits, not to enable appropriate lawsuits to go forward.” Mr. Lytton, the law professor, said the law "pretty much shut down attempts to use the court system in a broad way to try to push for greater industry self-regulation of marketing and distribution of firearms." But the exceptions have not put a complete halt to litigation. In 2008, a lawyer for the Brady Center to Prevent Gun Violence wrote that “reports of the demise of gun liability cases were premature,” adding that “well-pleaded, carefully crafted cases can still proceed against irresponsible gun companies.” While courts have barred claims in numerous instances because of the 2005 law, there have also been cases in which suits were allowed to proceed because of the exceptions, according to the Law Center to Prevent Gun Violence. How courts interpret the exceptions is still unfolding, and lawyers have tried to craft lawsuits that could meet their criteria. As one example, families of some of the victims of the 2012 mass shooting at Sandy Hook Elementary School in Newtown, Conn., are suing Remington Outdoor, the maker of the Bushmaster rifle used in the shooting, among other defendants. The families are arguing that the gun maker knew, or should have known, that selling assault rifles to civilians “posed an unreasonable and egregious risk of physical injury,” and that its conduct violated the Connecticut Unfair Trade Practices Act. Remington Outdoor is seeking the suit’s dismissal. The 2005 law, it said in court papers last month, “was enacted to protect firearm manufacturers against the very claims plaintiffs make in this case.” — Thomas Kaplan Close

She also used her visit to Waterloo to press a recurring theme of recent days, calling Mr. Sanders weak on gun control. She pointed to his vote in 2005, as a member of the House, for a bill limiting the liability of gun makers and sellers. She also embraced President Obama’s recent executive actions on guns.

The crowd at the Electric Park Ballroom in Waterloo cheered Mrs. Clinton’s arrival and gathered around her to take selfies after she spoke. However, many in the audience said that they remained undecided and that they were hoping that seeing her in person might help them choose.

“I have no dislike for her or anything like that,” said Floyd Bumper, 32, of Waterloo. “But she’s my second choice.”

Yamiche Alcindor contributed reporting from Pleasantville, Iowa.