By Julia Preston, New York Times

DENVER — Donald Trump’s harsh campaign rhetoric against Mexican immigrants has helped him win a substantial delegate lead in the Republican primary, but it is also mobilizing a different set of likely voters — six of them alone in the family of Hortensia Villegas.

A legal immigrant from Mexico, Villegas is a mother of two who has been living in the United States for nearly a decade but never felt compelled to become a citizen. But as Trump has surged toward the Republican nomination, Villegas — along with her sister, her parents and her husband’s parents — has joined a rush by many Latino immigrants to naturalize in time to vote in November.

“I want to vote so Donald Trump won’t win,” said Villegas, 32, one of several hundred legal residents, mostly Mexicans, who crowded one recent Saturday into a Denver union hall. Volunteers helped them fill out applications for citizenship, which this year are taking about five months for federal officials to approve. “He doesn’t like us,” she said.

Overall, naturalization applications increased by 11 percent in fiscal 2015 over the year before, and jumped 14 percent during the six months that ended in January, according to federal figures. The pace is picking up by the week, advocates say, and they estimate applications could approach 1 million in 2016, about 200,000 more than the average in recent years.

While naturalizations generally rise during presidential election years, Trump provided an extra boost this year. He kicked off his campaign in June describing Mexicans as drug-traffickers and rapists. His pledge to build a border wall and make Mexico pay for it has been a regular applause line. He has vowed to create a deportation force to expel the estimated 11 million immigrants here illegally, evoking mass roundups of the 1950s.

Among the 8.8 million legal residents eligible to naturalize, about 2.7 million are Mexicans, the largest national group, federal figures show. But after decades of low naturalization rates, only 36 percent of eligible Mexicans have become citizens, while 68 percent of all other immigrants have done so, according to the Pew Research Center.

“A lot of people are opening their eyes because of all the negative stuff Donald Trump has brought,” said Villegas’ husband, Miguel Garfío, 30, who was born and raised in Colorado and came to the workshop here to help his wife and other family members become citizens. His parents came from Mexico in the 1980s and worked hard all their lives, he said, helping him create a construction company in Denver that now employs 18 people. Contrary to Trump’s depiction, he said, none of his relatives have criminal records.

This year immigrants seeking to become citizens can find extra help from nonprofit groups and even from the White House. In September, President Barack Obama launched a national campaign to galvanize legal residents to take the step. They can now pay the fee, $680, with a credit card, and practice the civics test online. They can get applications at “citizenship corners” in public libraries in many states.

The White House recruited Fernando Valenzuela, the legendary Mexican-born pitcher who naturalized only last year, and José Andrés, the Spanish-American chef, to make encouraging advertisements and to turn up at swearing-in ceremonies. On Presidents’ Day, administration officials swore in more than 20,000 new citizens. On Wednesday the administration announced $10 million in grants to groups guiding immigrants through the process.

The majority of Latinos are Democrats, and some Republicans accuse the White House of leading a thinly veiled effort to expand the ranks of the president’s party. But administration officials argue the campaign is nonpartisan, noting that immigrants who become citizens improve their incomes and chances for homeownership.

“I certainly don’t care what party they register with, I just want them to become citizens,” said Leon Rodriguez, director of U.S. Citizenship and Immigration Services, the federal agency in charge of naturalizations.

Aside from Colorado, naturalization drives are taking place in Nevada and Florida, states likely to be fiercely contested in November where Latino voters could provide a crucial margin. One nonprofit group, the New Americans Campaign, plans to complete 1,500 applications at a session in Marlins Park baseball stadium in Miami on March 19.

Among the groups the White House is supporting are immigrant rights organizations and labor unions, which say their goal in holding dozens of citizenship workshops this spring is to build immigrant voting power. They want to boost support for legislation creating a path to citizenship for unauthorized immigrants, which Obama has long promised but has never been able to push through Congress. Recently naturalized immigrants, after all the effort they must make, are more likely to vote than longtime citizens.

“People who are eligible are really feeling the urgency to get out there,” said Tara Raghuveer, deputy director of the National Partnership for New Americans, a coalition that helped put on the workshop in Denver. “They are worried by the prospect that someone who is running for president has said hateful things.”

Trump says he is confident Latinos will support him, because he has employed many thousands of them over the years.

“I’m just telling you that I will do really well with Hispanics,” he said in the Republican debate in Houston on Feb. 26.

But in a poll of Latino voters on Feb. 25 by the Washington Post and Univision, the Spanish language television network, 80 percent had an unfavorable view of Trump, including 72 percent with a very unfavorable view, far more than for other Republican candidates.

Hope Hicks, a spokeswoman for the Trump campaign, said, “No one will benefit more from Mr. Trump’s pro-worker immigration reforms than the millions of immigrants who already call America home.” She said his proposals include “limiting the ability of corporations to replace them with new, lower-wage workers brought in from abroad.” Polls show Hispanic workers favor raising wages instead of importing foreigners, Hicks said, adding, “That is the core moral principle that will guide immigration policy in a Trump administration.”

Many Mexicans have been content to live in the United States with their resident green cards. The naturalization fee is high, and Mexicans often underestimate their English and worry they will fail the test, said Manuel Pastor, a sociology professor at the University of Southern California who studies citizenship. Many Mexicans have family members who are undocumented and think twice before engaging with the government, he said.

Yet many Mexicans joined a naturalization rush in 2007, when the threat of a fee increase right before the 2008 election prompted more than 1.3 million immigrants to apply. This year, no such increase looms. There is no hard deadline for immigrants hoping to vote in November, but with the agency approving naturalizations in about five months, immigrant groups are pressing to get applications in before May 1 to allow new citizens time to register to vote.

At the Denver workshop, many aspiring voters agreed on why they are naturalizing this year.

“Donald Trump never! Never!” said Minerva Guerrero Salazar, 40, who works for a uniform rental company after moving here from Mexico in 2002. “He has no conscience when he speaks of Latinos. And he is so rude. I don’t know what kind of education his mother gave him.”

Several women said they hoped to vote for Hillary Clinton, the Democratic front-runner.

At least one man liked a Republican. Dr. Oscar Arguello Rudín, 71, a Costa Rican who has been a resident since 1971 and recently retired as chief of surgery at a hospital in Colorado Springs, favored Gov. John Kasich of Ohio.

Mary Victorio, 22, a Mexican-born student at the University of Colorado, Denver, said she would vote Democratic but was grateful in one way to Trump. “He gave us that extra push we needed to get ready to vote, to prove to people who see us negatively they are wrong,” she said.

To Learn More:

Trump Anti-Immigrant Rhetoric Emboldens Hate Mongers (by Noel Brinkerhoff, AllGov)

Donald Trump Has a Plan for Deporting Millions of California Illegal Immigrants (by Ken Broder, AllGov California)