Perhaps the most difficult issue for the Republican Party is Mr. Trump’s call to deport not only all illegal immigrants, but also young people who came to this country as children and have received protections though executive actions by Mr. Obama.

“We have to keep the families together, but they have to go,” Mr. Trump said on “Meet the Press” in an interview, broadcast on Sunday, aboard his jet at the Des Moines airport.

It is an issue that other Republicans had tried to duck, focusing on securing the southern border. At the first Republican debate for the bottom seven candidates in the polls, former Gov. Rick Perry of Texas framed the issue typically, saying that only when the border is sealed tight will Washington be “up to a conversation to deal with the millions of people that are here illegally.”

A sizable core of Republicans favors such mass deportations, including 43 percent in a recent Wall Street Journal/NBC News poll. But that position is out of step with three-fourths of Americans — including 76 percent of independents — who believe illegal immigrants should be allowed to stay in the country if certain conditions are met, according to a Pew Research Center survey in June. And a narrow majority of Republicans favor giving undocumented immigrants a path to legal status or citizenship.

Representative Steve King of Iowa, one of Congress’s most hawkish Republicans on immigration, discounted concerns that the issues raised by Mr. Trump would hurt the party’s brand.

“They often say anytime you don’t pander to Hispanics, you cost Republican votes,” he said. “I think Trump gets stronger with this position that says enforce our immigration laws.”