Mr. Trump would need to expand the basket to include immigrants living in the United States illegally who have been charged but not convicted of crimes, those who have overstayed visas, those who have committed minor misdemeanors like traffic infractions, and those suspected of being gang members or drug dealers.

Targeting workers for immigration-related offenses, such as using a forged or stolen Social Security number or driver’s license, produced a significant uptick in deportations under Mr. Bush. But the practice was widely criticized for splitting up families, gutting businesses that relied on immigrant labor and taking aim at people who went to work every day, rather than dangerous criminals.

During Mr. Bush’s second term, deportations increased to 360,000 from 246,000, while the number of immigrants convicted of crimes who were deported remained virtually stagnant, according to government statistics. And a 2007 report by the National Council of La Raza and the Urban Institute that analyzed work site raids in Massachusetts, Colorado and Nebraska said a majority of the children affected by the arrest and deportation of their parents were United States citizens and were infants, toddlers or preschoolers.

In 2009, the Supreme Court ruled unanimously that federal identity-theft laws may not be used against illegal workers who used fake Social Security numbers to get jobs, unless those workers knowingly used numbers that belonged to real people.

“This is the low-hanging fruit in the system: nondangerous undocumented immigrants with families,” said John Sandweg, a former acting director of ICE under Mr. Obama. “Those people don’t hide. Criminals hide.”

Michael Chertoff, the Homeland Security secretary under Mr. Bush, defended workplace raids. He said these sweeps proved a potent way to protect workers by tamping down on the “ecosystem of smuggling,” which emboldens migrants to sneak into the country and empowers employers to hire them to work illegally, in unregulated and often inhumane conditions.

“We found it to be effective in a targeted way,” Mr. Chertoff said in an interview. “We didn’t willy-nilly stop at a workplace and raid it.”