Dora Eglis Ramírez and Pavel Brigido Rivero set out from Cuba to seek asylum in the United States last year, as the coronavirus rampaged across Latin America.

Starting their trek in Guyana, they managed to cross eight countries, sleeping in buses and doing odd jobs, without ever contracting the virus.

Then they crossed the border into the United States.

U.S. Border Patrol agents intercepted them late last month in Southern California and transported the couple to a heavily crowded border station. They spent 10 days and nights in cells crammed with Brazilians, Cubans, Ecuadoreans and Indians.

Mr. Rivero, 45, came down with the coronavirus and spent the next two weeks isolated, along with his still-healthy wife, at a hotel with about 200 other migrants who had tested positive for the virus or had been exposed to someone who did.

“I was healthy until I got locked up,” he said.

As the United States vaccinates larger numbers of people and several states begin to reopen after seeing lower infection rates, the failure of U.S. authorities to test adult migrants for the coronavirus in jam-packed border processing centers is creating a potential for new transmissions, public health officials and shelter operators warn, even among migrants who may have arrived healthy at America’s door.

More than 170,000 migrants crossed the border in March — many coming from countries still grappling with high infection rates — but the Border Patrol is conducting no testing for the coronavirus during the several days that the newly arrived migrants are in U.S. custody except in cases where migrants show obvious symptoms.

The government says it has insufficient time and space to test migrants upon their arrival. So while migrants get a basic health screening, testing is being postponed until their release to local community groups, cities and counties, usually after the new arrivals have spent days confined in tight spaces with scores of strangers, often sleeping shoulder to shoulder on mats on the floor.

Unaccompanied children are being tested, but only after they have spent around three days in custody, just before being loaded onto buses or planes for transport to government-run shelters.

U.S. officials say the challenges to testing all the new arrivals when they are first apprehended are insurmountable. There have been no instances of mass spread at U.S. border facilities, and overall numbers of cases are relatively low, according to the Department of Homeland Security. About 5 percent of all single adults and families tested after their release since March showed a positive result, according to the agency, while among the thousands of unaccompanied minors now in custody, the rate has been about 12 percent.

But local officials and shelter operators said they feared that the actual number of infections could be much higher.

In California’s Imperial Valley, where the Cuban couple was apprehended, 15 percent of the migrants released by the Border Patrol into the community between April 7 and April 13 tested positive for the virus — three times higher than the official average, according to the California Department of Social Services.

And the operator of several large shelters where migrant children go after their release from border processing said one out of five children at those facilities was showing a positive test result on arrival.

“In theory, those who test positive could have infected other people before arriving here,” said Diego Piña Lopez, the program manager at Casa Alitas, a respite center for migrant families in Tucson, Ariz. Staff members there have been performing rapid coronavirus tests on dozens of migrant families each day after their release by the Border Patrol.

Migrants who have a positive result are transferred to a shelter operated by the city. Others spend a night or two at the respite center and then board planes or buses to their destinations around the United States. Some of them could well have infections contracted in Border Patrol facilities that did not register on tests during the brief time they spent at the respite center, immigrant advocates warned, and could unknowingly expose others as they travel to join friends and family elsewhere in the country.

“People who were on the bus or in the cell with people who tested positive are going to test positive,” said Mark Lane, who runs a small humanitarian organization in San Diego, the Minority Humanitarian Foundation. “Uber drivers, taxi drivers and people like us, people who are not fully vaccinated, are getting exposed. Today I took two guys who were released and put them in a T.S.A. line with 500 people on it.”

John Modlin, the interim Border Patrol chief for the Tucson sector, said it took 90 minutes to three hours to process each migrant, including fingerprinting, gathering personal information and running a background check. Testing for the coronavirus and waiting for results would add another 20 minutes, he said.

“That’s 20 minutes times a thousand people,” Mr. Modlin said. “The Border Patrol does not want to get in the business of testing or inoculating people.”

Dr. Pritesh Gandhi, the chief medical officer at the Department of Homeland Security, said that “operational limitations” have precluded doing virus testing “on the front end,” but that medical teams are working intensely with nonprofit groups and local officials to make sure migrants are screened immediately and tested later, a strategy that he said was starting to show results with fewer people getting sick.

“At the earliest possible moment we can do something about it, we test,” he said in an interview. “And so there are limitations. The question that any public health operator has to ask is, ‘What is the earliest point you can effect change?’”

Some cities and counties have balked at having to conduct the bulk of all coronavirus testing for adult migrants. In El Paso, the county judge, the local Catholic bishop and other community leaders sent a letter to the secretary of homeland security, Alejandro Mayorkas, arguing that testing migrants was “beyond the capacity of the combined efforts of our local governments and N.G.O. community.”

The mayor in Yuma, Ariz., Douglas J. Nicholls, said that before the local medical center took over testing, migrants were being dropped off by the immigration agents at the side of a road or in parking lots — with no testing for the coronavirus.

“It’s completely crazy,” Mr. Nicholls said. “It’s not the way we should be handling things during a pandemic.”

Last week, the Texas attorney general, Ken Paxton, filed a lawsuit alleging that the federal government was “encouraging the spread of Covid-19 at the border” by keeping potentially infected migrants housed closely together in government custody.

Mr. Paxton said in a statement that President Biden was demonstrating “outright disregard of the public health crisis” by “welcoming and encouraging mass gatherings” of migrants in border facilities.

In a few cities, a contractor hired by Immigration and Customs Enforcement has begun to conduct coronavirus testing of migrants after their release from the Border Patrol and is arranging isolation space in hotels for those who test positive.

The Biden administration has continued to expel many who have entered the country without authorization, using a public health emergency law initially invoked by former President Donald J. Trump.

But the government of Mexico has refused to take back families traveling with children under the age of 7 along large stretches of the border with Texas. It has also rejected returns of migrants from outside Central America, who represent a growing number of crossers — many of them from Ecuador and Brazil, countries still hit hard by the coronavirus.

Migrants themselves are expressing worry about spending so much time in close quarters after being apprehended by U.S. authorities.

Jemerson Kener, a Brazilian who crossed the California border last month, tested positive for the coronavirus after spending four days at a crowded Border Patrol station.

“In a pavilion meant for about 20, there must have been 100 men,” he said.

Once he was told he had the virus, he was sent to a hotel in Holtville, Calif., where he said about 100 Brazilians were isolated, along with infected Cubans, Ecuadoreans and migrants from several Asian countries.

“I got really sick. Jesus, my throat was killing me,” said Mr. Kener, 33, who received medicine from a nonprofit group that is running the isolation operation at the hotel.

On April 12, after testing negative, he was allowed to head to Maryland, where he said a job in construction awaited him.

Cindy Mendez, a Honduran girl who crossed the border in February to join her mother in the United States, said she tested positive for the coronavirus after being housed for two weeks in a processing center in Donna, Texas, that in March was operating at more than 700 percent of the capacity it was designed for.

“We were sleeping on the ground on top of each other,” she said.

Department of Homeland Security officials stressed that there were no facilities for testing efforts at Border Patrol processing stations, particularly for children, who have to be separated by gender and age. Children are now traveling to shelters in separate buses depending on their Covid-19 status, an improvement from past months.

The agency’s focus has been on moving migrants out of custody faster, which is key to lowering their exposure, and the strategy has been successful: Data released on Tuesday showed that the number of unaccompanied minors in custody had dropped 80 percent in the past month.

But even tracking those migrants who have the coronavirus can be tough.

Andrea Rudnik, whose nonprofit organization, Team Brownsville, provides aid to a hotel for coronavirus isolation in Brownsville, Texas, said many migrants who tested positive had disappeared before their mandatory separation period was up.

“They want to leave,” she said, “and if they realize, ‘Hey, I can just take a taxi from this hotel back to the bus station and get out of there,’ then they’ll do it.”