President Biden, under pressure to address a surge of migrants who are overwhelming resources in New York and several other American cities, is ramping up efforts to ensure that some of those people can get jobs — a move designed to ease the spiraling political and financial costs of the crisis.

After more than two years in which his administration has struggled to find short-term solutions to an immigration system that has been badly broken for decades, Mr. Biden now faces demands from within his own party to confront the consequences of that migration, hundreds of miles from the border with Mexico.

Under federal law, migrants have to wait about six months after they file their asylum applications to apply for permission to work in the United States. The requirement has vexed cities like New York, where thousands of people have taken refuge in shelters, straining the system.

Some people who come into the United States legally through special programs are able to request work permits immediately, but they do not always do so. The Biden administration is focusing on those people in a new campaign.

In the last week, federal officials have sent more than one million text messages to migrants across the country who are eligible for permits to work in the United States but have not yet applied for them. Officials say the messages, which are sent in English, Spanish, Haitian Creole, Russian and other languages, are a “first-of-its-kind national campaign” by the federal government.

In addition, they have distributed fliers with QR codes to be posted at residential facilities, shelters, legal services clinics and other public places where new arrivals gather. Migrants can scan the codes with their phones to get information about how to download work authorization documents.

Administration officials said they did not have the official number of migrants in the country who are eligible for work permits but have not applied. But they said the government has distributed close to $770 million in grants to localities, including about $140 million to New York, to bolster services for the migrants. The administration has asked Congress for another $600 million in supplemental funding for this year, and $800 million for next year.

The efforts are a direct response to angry frustration voiced by Democratic leaders like Kathy Hochul, the governor of New York, and the mayors of some of the country’s biggest cities. Eric Adams, the mayor of New York City, declared on Thursday that the cost of caring for more than 100,000 new arrivals could reach $12 billion over three years and “will destroy New York City.”

The issue of how to deal with migrants, and who should shoulder the cost of caring for them in the short term, has led to a fierce and public intraparty spat in the midst of Mr. Biden’s re-election campaign.

Local officials have said the funding and the steps to allow migrants to work do not go far enough. And Republicans who backed former President Donald J. Trump’s tough immigration measures are now seizing on the comments from top Democrats, happily quoting those in New York who say the Biden administration must do more to secure the border.

In response, Mr. Biden and his top aides say Republicans are to blame for refusing to even consider proposals that the president made on his first day in office for a broad overhaul of the nation’s immigration laws. Conservative Republicans have blocked bipartisan efforts to modernize the system for decades.

The political repercussions of a continued migrant crisis have heightened tensions inside the Biden administration. According to emails reviewed by The New York Times, officials have worried for months about the impact of increased migration on big cities, where many migrants head once they cross the border.

The decision by two Republican governors, Greg Abbott of Texas and Ron DeSantis of Florida, to ship thousands of migrants on buses to New York, the District of Columbia and other locations added to the concern and led the local and state officials — all Democrats — to publicly air their anger over the situation.

One key demand from Ms. Hochul and Mr. Adams in recent weeks has been for the federal government to take actions to allow more of the immigrants to work in the United States. That would allow them to leave public shelters and provide for themselves.

Ms. Hochul made that case to Jeff Zients, the White House chief of staff, during a meeting last week. Afterward, the White House released a statement saying that “the administration will work with New York State and New York City on a month of action to help close the gap between noncitizens who are eligible for work authorization and those who have applied, to meet labor needs in New York.”

The issue has been building for months.

When Mr. Biden announced in January a new program that would open the borders to 30,000 migrants each month from Cuba, Haiti, Nicaragua and Venezuela, he hailed it as a humane and orderly way to limit surges in migration and prevent dangerous overcrowding at the border with Mexico.

But his administration did not anticipate that many of those migrants would not immediately apply for work permits despite being eligible, leaving tens of thousands of people seeking housing, medical care and other costly support from the cities they settled in across the United States.

Only last week — under pressure from New York, where many of the migrants were bused by Republican governors — did the administration launch a concerted effort to get those migrants to sign up for permission to work.

Administration officials said Friday that getting more of the recent migrants to seek work authorization was unlikely to solve the entire problem in New York, where local laws require the city government to provide housing for any migrant who needs it, even if it means putting them up in costly hotels.

The so-called right-to-shelter law in New York is a powerful “pull factor,” administration officials said, attracting migrants who are otherwise unsure where to go after they get to the United States. Officials at the White House and the Department of Homeland Security say they admire the humanity of the law but acknowledge that it adds to the city’s burden in a way that the federal government cannot control.

The effort to deal with the needs of migrants after they spread out to cities is also hampered by a lack of information about who they are and where they go, officials said.

People who enter the United States often tell the authorities where they intend to go after leaving the border. But that is often only their first stop, and they do not stay there. D.H.S. officials said they did not know exactly how many of the 110,000 migrants who arrived in New York City in the last year were currently eligible for work permits.

Some might have come into the United States legally, through a program that allows them to stay in the country for up to two years if they are fleeing from Cuba, Haiti, Nicaragua or Venezuela. Others might have used a new Customs and Border Protection app to make appointments at the border, which also gives them the right to work.

But others who found their way to New York City may be what the border authorities call “gotaways,” people who slipped by the authorities at the border and made their way to their final destination without being caught. Those people would not be eligible for work permits because they entered the United States illegally.

Migrants who came to the United States illegally and have since started the process to claim asylum are required by law to wait 150 days before applying for a work permit. It takes a minimum of another 30 days before they can begin to work, and the current backlog in processing time sometimes takes weeks longer.

Individuals who are close to the 150-day threshold are among those being targeted by the text messages and QR codes, officials said.

Eileen Sullivan contributed reporting.