Is the immigration crackdown already showing up in the labour market?

Employment growth in industries that rely heavily on unauthorized workers has slowed. There has been a large decline in the foreign-born labor force since March. And recent immigrants appear more reluctant to take part in the Labor Department’s monthly survey of households.

Still, there is no definitive measure of the unauthorized workforce, and the data that does exist can be volatile and difficult to parse.

The Labor Department relies on two surveys to produce its monthly jobs report—one of employers, the other of households—and each has blind spots.

The employer survey is where the main monthly jobs number comes from. It queries firms on how many workers they have on their payrolls. It doesn’t ask questions about workers’ legal status, but some companies, such as those with people working off the books, probably don’t respond accurately. The employer survey also doesn’t include farmworkers and workers in private households—two employment categories with a large number of unauthorized workers.

That survey does include some industries with large shares of unauthorized workers, such as landscaping services and slaughterhouses. And as a group, these have been adding jobs at a slower rate than other private-sector employers since last summer, when the number of people coming into the U.S. began to fall, according to an analysis by economist Jed Kolko, who was Under Secretary for Economic Affairs at the Commerce Department during the Biden administration. But that slowdown has actually been less pronounced in recent months, according to Kolko.

“We probably haven’t seen the full effect from immigration policies this year yet," he said.

The Labor Department’s separate survey of households is where the unemployment rate, among other labor measures, comes from. It doesn’t ask questions about immigration status, though it does ask if people were born in the U.S. or not. It also makes a pledge of confidentiality, but many unauthorized immigrants are still probably reluctant to participate in the survey, and lately that reluctance might have grown. Indeed, the response rate of recent immigrants has fallen this year.

A man under arrest by federal law-enforcement agents in Georgia earlier this year.

That, Goldman Sachs said in a note to clients last week, raises concerns “that the survey may have missed many unauthorized immigrants who are scared to go to work amid the intense immigration crackdown."

Data released alongside Friday’s jobs report showed that the number of foreign-born people either working or looking for work fell by about one million from March to May. That was the biggest two-month decline in the foreign-born labor force since the early days of the pandemic. Some economists pointed to that decline as an indication that more unauthorized workers are exiting the labor force.

That data is extremely volatile, though, and not adjusted for seasonal swings, which are severe in sectors such as farming. The ratios of both foreign- and native-born employed people to population have behaved similarly in the past year.

Immigration has long been an important source of labor in the U.S., particularly in recent years as the number of unauthorized immigrants flowing across the border surged. But the number of people coming into the country began falling in the later part of last year after the Biden administration tightened up the border. And it has fallen precipitously since President Trump came back into office, while raids on workplaces have made some immigrants fearful of showing up for work.

Workforce authorization backlogs continue to grow and reports of arrests at regular immigration appointments rise, scaring some potential applicants away from the process.

Nicole Hallett, who runs the Immigrants’ Rights Clinic at the University of Chicago, which offers free legal assistance, said many of her clients have quietly exited the workforce as new Trump administration policies put pressure on their ability to work legally in the U.S. Work permits also often create a path for migrants to other identification and access other social programs.

That said, many of the people who crossed the Southern border last year are likely still entering the labor-force pipeline. Many unauthorized immigrants, such as those who apply for asylum, are eligible for work permits, but often are required to wait before applying, and then wait for approval. As of April—the latest data available—the number of people applying for these permits was high, points out Wendy Edelberg, an economist at the Brookings Institution.

Hallett said she believes that many people are likely still applying for work permits even now. Workers have to feed their families, after all.

“Work authorization is so valuable that I think people are still applying," Hallett said. “But I would say that the level of fear that people are feeling, just because they know that any time they file something, they get put on the radar, is a lot higher."

New policies that strip work permit eligibility will likely have a harsher impact on workers with better jobs, not those who have found a way to survive by being paid under the table, Hallett said. An employer who pays attention to immigration law is more likely to also care about other labor laws, she has found.

Conchita Cruz, co-executive director of the Asylum Seeker Advocacy Project, which represents more than 700,000 migrants in all 50 states, said the group’s members aren’t the only ones who feel the moment’s uncertainty.

“Employers want the answers just as bad as the workers do in this case," Cruz, an attorney, said. “It’s a rare instance where business owners and the business community are on the same page as unions and the labor movement and immigrants."

The group’s members span a wide range of industries, and for example include both doctors and cleaners at the hospital, she said.

Immigrant workers are playing a critical role in many different industries, Cruz said, and employers don’t want to lose them. “At the end of the day, it’s very costly to have to retrain someone, to have to find new employees, to have to replace people who otherwise are doing a great job and you would want to keep in a job."

It will still take time for the effects of the new immigration policies to show up. UBS economists further note that the unemployment rate for workers ages 16 to 19 rose to 13.4% in May from 13% a month earlier. Because teens typically work in sectors with higher shares of immigrant workers, the economists expect teen unemployment would fall, not rise, in response to immigrants leaving the labor force.

It might not take long for the immigration crackdown to show up more clearly. The Labor Department’s employer and household surveys are based on midmonth readings. That means that the May jobs report that was released last week didn’t include the dialing up of immigration enforcement that came in late May after top White House aide Stephen Miller told Immigration and Customs Enforcement to “just go out there and arrest illegal aliens."

The June jobs figures, due July 3, will be based on the pay period and week that includes the 12th of this month—that is, Thursday.

Write to Justin Lahart at Justin.Lahart@wsj.com