CHICAGO — Bulging suitcases lined the wall of windows facing the street at a North Side police station in Chicago one morning this week. Air mattresses, blankets and pillows covered the shiny lobby floor. And more than 40 women, men and children were crowded together wherever they could find space — sleeping, chatting in Spanish or eating forkfuls of scrambled eggs and sausages that a volunteer had arranged on a long table.

In the final days of Mayor Lori Lightfoot’s tenure, her administration has been confounded by a sudden surge of migrants, mostly from South America, who have been bused or flown from southern states to Chicago Union Station or O’Hare International Airport, and then brought to police stations like the one on the North Side to await shelter beds.

City officials and volunteers say the response to the influx has been fractured, as the chaotic reality of the migrant crisis in the United States strains the Democratic-run city’s resources for housing and feeding thousands of new arrivals. And the influx is expected only to increase after a change in federal rules takes effect on Thursday.

For Chicago, the challenge is not merely a practical one, but also a test of the city’s own sense of identity. Since the late 19th century, when Jane Addams established Hull-House, a social settlement that drew migrants, Chicago has seen itself as an entry point for newcomers, a sanctuary city that welcomes people from the outside.

“We’re not living up to it right now,” said Mary Kay McDermott, a volunteer with a local refugee resettlement group, as she surveyed the scene at the police station, one of dozens of stations where migrants have been given temporary shelter. “We say we’re a sanctuary city,” Ms. McDermott said, “but I don’t think we’ve put in place the services to deal with this.”

As the number of migrants entering Chicago has significantly increased in recent weeks, their presence around the city — in police stations, in park facilities and on neighborhood streets — has become far more visible. More than 100 new migrants are arriving each day now, officials said, compared with about a dozen a day a few months ago.

On Tuesday, Ms. Lightfoot declared a state of emergency. She said that Greg Abbott, the Republican governor of Texas who has bused thousands of migrants out of Texas to Democratic-held cities far from the border, had resumed sending migrants to Chicago in the midst of a “national humanitarian crisis.”

“We should all understand that this crisis will likely deepen before we see it get better,” said Ms. Lightfoot, a Democrat, adding that “through a unified effort in accordance with its values as a welcoming city, Chicago is doing everything it can to respond to the urgency of this matter.”

Even more migrants are expected to reach cities like Chicago after Title 42, a federal pandemic restriction that allowed the swift expulsion of many migrants at the southern border, expires on Thursday.

Cities including New York and Washington have seen influxes begin even before Title 42 ends. More than 8,000 migrants have come to Chicago in the past year, and by Wednesday, the city’s shelter beds were full and officials were scrambling to find more space.

The city has set up at least 10 shelters for housing migrants, working with community organizations to provide transportation and other services. The cost of handling the influx of migrants from January to June is close to $125 million, a city official said; the state of Illinois has approved $30 million to aid the city’s response.

The migrant influx is coming at a critical moment of transition for Chicago: Brandon Johnson, the mayor-elect, will be sworn in on Monday to succeed Ms. Lightfoot, and will inherit a problem that is becoming more urgent each day.

Police stations, where homeless people are allowed to wait before they are placed in a shelter, are now serving as the city’s front step for migrants. Police officials have grown alarmed at the volume of people who have taken up residence on the floors of police stations — in at least one location, filling nearly the whole lobby and forcing police officers to send neighborhood residents seeking help with routine police matters to other precincts.

Some of the migrants, including pregnant women and small children, are ill with colds, pink eye or Covid-19, leaving volunteers and city workers scrambling to find them health services.

Thousands of migrants have come to Chicago in the last year with few belongings, limited English skills and almost no idea what lies ahead for them as they seek asylum and permanent residence. While the city seeks more places to house migrants, officials have used hotels, park district buildings and vacant schools as makeshift shelters.

“It’s unsustainable,” said Maria Hadden, a City Council member who represents Rogers Park on the Far North Side.

Ms. Hadden said that as summer approaches and city parks prepare for summer sports and camps, the city may be forced to use facilities intended for other purposes to house asylum seekers instead.

“Whose park district is going to close down and move programming?” she said. “Which school is not going to be able to do summer programming? Are we going to have to cancel some things at Navy Pier?”

The migrants who have arrived in the city are not trying to leave, said Dr. Evelyn Figueroa, a family medicine doctor who is working with the city on handling the arrivals. Chicago is already home to a large Spanish-speaking population, and close to 30 percent of the city’s residents are Hispanic.

“They want to live here,” Dr. Figueroa said. “They want to work here. They like Chicago.”

Outside one temporary shelter, Anthony Piña, 33, said he had arrived in Chicago the day before, and hoped that he would find more job opportunities, ideally in the construction business, than he had in his native Venezuela. “It was very tough,” he said of life there, adding that he would work for months and earn only the equivalent of $50.

In many Chicago neighborhoods, a robust response from volunteers, churches and nonprofit organizations has helped migrant families find temporary apartments, clothing, and social services. Some volunteers have opened their homes to migrant families so that they can shower or rest. At police stations, officers have brought in food they have prepared for the migrants; one desk sergeant, worried that the families were not sleeping comfortably on the floor, brought in pillows.

But in some neighborhoods, the Lightfoot administration’s scramble to open large shelters for migrant families has been met with open hostility.

In the South Shore neighborhood on the South Side, hundreds of residents gathered in a school auditorium last week to hear about the city’s plans to repurpose a former high school in the area into a shelter for migrants, most of them from Venezuela.

Nubia Willman, a City Hall official, was loudly booed and taunted as she faced the crowd from the stage. Residents lined up in front of a microphone and, for two hours, implored city officials to reverse course and put the shelter in another part of the city.

Their neighborhood is already strained for resources, they said, and there are not enough police officers to respond to crime and other problems that already exist.

“When I heard about the immigrants were going to be put in a shelter here, my first thought was, ‘Are we going to be safe?’” said one resident, Patricia King.

Clifton Bradley, the owner of a media production company, told officials that he was upset that the migrants were allowed to enter the United States to begin with.

“What I would like to say is, the sanctuary city is wrong,” he said, drawing applause from the crowd. “No one is entitled to anything.”

In an interview, Ms. Willman said that cities are not typically responsible for resettling asylum seekers, and that Chicago was suffering from a lack of federal funding.

“We’ve never had to resettle people in this way — it’s a new situation for us,” she said. “Chicago calls itself a sanctuary city, but I think we all need to agree on the definition of what a sanctuary city is. Is it that we just don’t call ICE when someone has entered the country through illegal means? Does it mean that we all agree that a sanctuary includes food, housing, clothing and warmth?”

Critics of Ms. Lightfoot, who was defeated for re-election in February after one term, said that she has had months to prepare for the migrant influx but failed to properly respond.

“I think this is characteristic of her administration,” said Byron Sigcho-Lopez, a City Council member who represents a ward on the West Side. “The lack of coordination, the lack of consultation with local communities and a total dismissal of the facts.”

At a city beach field house along Lake Michigan, dozens of migrants have been staying temporarily in space that is used in the summer for storage and for lifeguards to take breaks. Many migrants there had only a blanket between them and the hard gym floor, and toddlers scampered around barefoot.

Outside the field house, a group of migrants accepted slices of quiche that a volunteer, Mary Elking, had prepared at home and brought for the group.

Jose Moran, 42, who was staying at the field house, said he had previously spent six days sleeping at a police station, after a journey from Venezuela to Chicago that had taken months. He was still adjusting to his new surroundings: the cool late-spring weather, the surprising vastness of one of the Great Lakes a few yards away.

“We’re tired, we’re exhausted,” he said. “We just want to stay.”