LOS ANGELES — Maria Elena Hernandez recently retrieved a flowery box tucked in her closet and dusted it off. For more than a decade, she has used it to store tax returns, lease agreements and other documents that she has collected to prove her family’s long years of residence in the United States.

“We have been waiting for the day when we can apply for legal status. In this box is, hopefully, all the evidence we’ll need,” said Ms. Hernandez, 55, an undocumented immigrant from Mexico who arrived in this country with three small children in 2000.

She had just learned of President Biden’s plan to offer a pathway to U.S. citizenship for nearly 11 million undocumented people, announced as part of a sweeping proposal to overhaul the nation’s immigration system.

The bill would allow undocumented immigrants who were in the United States before Jan. 1 to apply for temporary legal status after passing background checks and paying taxes. As newly minted “lawful prospective immigrants,” they would be authorized to work, join the military and travel without fear of deportation. After five years, they could apply for green cards.

The president’s proposal would be perhaps the most ambitious immigration redesign passed since 1986, when President Ronald Reagan signed the Immigration Reform and Control Act, which legalized three million people.

Converting more than three times that many people into full citizens could open the door to one of the most significant demographic shifts in modern U.S. history, lifting millions out of the shadows and potentially into higher-paying jobs, providing them with welfare benefits, health coverage and Social Security eligibility while eventually creating many new voters.

“This is the boldest immigration agenda any administration has put forward in generations,” said Muzaffar Chishti, senior fellow at the Migration Policy Institute. “But given that the Democrats have razor-thin majorities in Congress, the administration needs to have its expectations tempered.” Legalizing just one group at first — say, farmworkers — might be “more realistic,” he said.

In a sign of the hurdles ahead, another one of Mr. Biden’s early immigration initiatives, a 100-day freeze on deportations, was temporarily blocked by a federal judge on Tuesday after a lawsuit by the Texas attorney general, an advocate of the Trump administration’s immigration crackdown.

Immigration reform has stalled in Congress time and again, primarily over what is widely known as amnesty. Despite beefed-up border enforcement and employer sanctions, Mr. Reagan’s overhaul failed to curb the arrival of unauthorized immigrants.

While Congress has wrestled with how to revamp the immigration system, the immigrants have continued to live, work and raise families in the United States. More than 60 percent have resided in the country for more than a decade and they have more than four million U.S.-born children. They account for 5 percent of the work force, representing the backbone of the agriculture, construction and hospitality sectors.

Many came for economic reasons. Others were fleeing political upheaval and violence. And about four in 10 did not enter the United States through the southwestern border. Rather they are visa overstays — tourists, students or temporary skilled workers who never left.

The family of Denise Panaligan, 27, came to the United States from the Philippines in 2002 after her father, a financial analyst, obtained an H-1B visa. They remained after their temporary legal status ran out.

“When people look at us, they don’t think undocumented because we are Asians,” said Ms. Panaligan, a middle school teacher in Los Angeles. “Coming in contact with deportation officers and ICE, we never experienced that. We’re invisible to enforcement.”

But their decision led to other indignities, she said. Her mother had to work as a housekeeper despite having a college education. Denise’s father, Art, died two years ago of brain cancer without ever returning to his homeland to see his own parents because he would not have been allowed back into the United States.

“The Biden plan would fulfill our hope of keeping the family together,” said Ms. Panaligan, who has two younger sisters, one of them born in the United States.

The largest share of unauthorized immigrants is from Mexico. Having survived treacherous river and desert crossings to reach the United States, they found a nation willing to look past laws that prohibit hiring them, to employ them in fields and factories, and in homes as nannies and housekeepers.

Maribel Ramirez and Eusebio Gomez of Mexico have toiled in California’s vineyards since crossing the border 19 years ago. They have managed to buy a home and raise two American-born sons. Their oldest, Eusebio Jr., 17, plans to enlist in the Marines. But Ms. Ramirez said she asked herself, “Why should my son give his life to a country that doesn’t value his parents?”

Presidents George W. Bush and Barack Obama both championed comprehensive immigration reform with a strong enforcement component and a pathway to legalization for undocumented people. But the immigration packages that were debated in Congress — in 2006, 2007 and 2013 — all stalled.

Among the concerns raised by opponents are that new citizens will vote as a solid Democratic bloc, displace American workers and become a burden on the health care system and other public services. Some predict that any legalization program would encourage more people from impoverished Latin American countries to make the trek north.

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“Legalizing countless millions of illegal aliens — even discussing it — rings the bell for millions more to illegally enter the U.S. to await their green card, and this cycle does not end,” said Lora Ries, a senior research fellow at the Heritage Foundation, a conservative think tank in Washington, and a former acting deputy chief of staff in the Department of Homeland Security.

Other experts argue there are benefits of legalization. Opening a pathway to citizenship for nearly 11 million people, seven to eight million of whom participate in the labor force, is tantamount to “an economic stimulus,” according to Giovanni Peri, an economics professor at the University of California, Davis.

Between 2005 and 2015, new immigrants accounted for nearly half of the growth in the working-age population, and in the next two decades, immigrants will be key to offsetting an aging population that is retiring. Demographers say the increased educational attainment of Americans coupled with a shortage of blue-collar workers highlights the need for immigrants, in ever larger numbers, to perform low-skilled jobs. About five million of them work in jobs designated as “essential” by the government.

Among the biggest backers of the Biden initiative are employers who rely on immigrants. Through the years, meatpacking plants, dairy farms and a multitude of other worksites have been caught up in immigration raids targeting unauthorized workers.

The Reagan-era amnesty in 1986 caused only a temporary drop in the number of undocumented immigrants because it was not accompanied by a robust system for legally bringing in low-skilled workers. Employers faced fines for knowingly hiring undocumented immigrants but were not responsible for vetting documents presented by job applicants, spawning a huge industry of fake Social Security numbers.

“The principle is simple: If you carry out a broad legalization, it doesn’t freeze undocumented migration flows as long as labor demand persists,” said Wayne Cornelius, director emeritus of the Center for Comparative Immigration Studies at the University of California, San Diego. “You need to increase the number of legal entry opportunities for future migrants.”

The illegal influx began to swell again in the early 1990s.

“Migrants arriving after 1986 would have vastly preferred coming legally, not having to pay hundreds of dollars to purchase false docs,” Mr. Cornelius said. “But not enough legal entry tickets were available.”

Economic imperatives prevailed, driving illegal immigration up year after year.

A building boom in Sun Belt states like Georgia, North Carolina and Arizona drew hundreds of thousands of undocumented construction workers. And as farm workers who had benefited from the amnesty aged and exited the fields, young undocumented labor arrived to replace them.

From 1986 to 2008 the country’s undocumented population swelled from three million to 12 million despite an exponential increase in funding for border security, including boots on the ground. The militarization did not reduce illegal entries. Instead, it turned a seasonal migration of mainly men who returned home each year to Mexico into a settled population of families.

The economic recession between 2007 and 2009 ultimately reduced the flow of immigrants. Despite successive waves of Central American migration, illegal entries remain substantially lower than in the early 2000s.

It has been about two decades since Ms. Hernandez stuffed a change of clothes in a bag, grabbed her three young children and closed the door on their lives in Jalisco, Mexico. Days later, they slipped across the border.

“We didn’t have papers, but I was determined to head north for a better life,” she said. “I had relatives who had benefited from a 1986 amnesty and I figured our day would come.”

Ms. Hernandez got a job in Los Angeles packing CDs. Her husband, Pedro Hernandez, joined them shortly after and found work at a nursery. Their children, Luis, then 7, and Elitania and Juan, then 4-year-old twins, thrived in school. They lived modestly, in apartments where the children slept in the bedrooms and the couple in the living room.

Their hopes were lifted, dashed and revived as Congress took up various immigration bills.

“Every president gave us hope that something good was coming, and then nothing. Again maybe. Then nothing,” Ms. Hernandez recalled.

Ms. Hernandez said she and her husband worked hard, paid their taxes and never got into debt.

“We tried to do everything right, according to the law, in a country that had not opened its doors to us, but that we had entered anyway,” she said.

Things got more challenging when Mr. Hernandez lost his job in 2006 His employer, concerned about government audits, had asked him to prove he was in the country legally.

When Immigration and Customs Enforcement agents began showing up in their neighborhood, the family avoided going out for days at a time. Mr. Hernandez rode his bike to a new job, figuring he would be less likely to encounter law enforcement than while driving his old Mazda. Ms. Hernandez signed up for adult English classes. As the children got older, she advised them to always be on their best behavior.

“I grew up hearing that you have to be an extra good citizen,” said Luis Hernandez, her oldest child, now 28. “Mom always said that: ‘People who have papers can make mistakes, but you can’t,’” he said.

Because of their undocumented status, Luis, who played club soccer, and Juan, a varsity football player, skipped games that required travel.

When Mr. Obama created a program known as DACA in 2012 to temporarily shield undocumented immigrants who had been brought into the country as children, Luis Hernandez and his siblings, then teenagers, applied immediately. It enabled them to work legally. They got their driver’s licenses.

But President Donald J. Trump’s incendiary anti-immigrant rhetoric and his administration’s attempt to rescind DACA generated deep anxiety in the Hernandez household. When Mr. Biden took office, the family rejoiced.

After the inauguration, Ms. Hernandez was at her dining room table thinking about the imminent birth of her second grandchild, who will be an American citizen. She and her husband planned to drive to Utah to meet the baby, and she worried about making a trip across state lines without legal status, lest law enforcement stop them.

When she learned that the president had unveiled a blueprint for legalization, she said, she was stunned at first. Then she went to retrieve the box of documents.

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