Immigrants themselves support Donald Trump’s call for a stiffer ideological test for newcomers to the U.S., according to a surprising new poll that suggests the GOP presidential nominee’s plans strike a chord.

More than 60 percent of immigrants who are registered voters back Mr. Trump’s call to make sure visitors and new immigrants share American values before they are admitted. That’s about the same level of support among all voters, according to Morning Consult, an analysis firm that’s been polling heavily on the election.

As for Mr. Trump’s plan to ban immigrants from countries with connections to terrorism, there was still majority support from immigrants, though less than for the ideology test. Some 52 percent of immigrant voters backed the ban, while 39 percent opposed it.

First-generation Americans — the children of immigrants — were far more enthusiastic, with 68 percent supporting it, the analysis firm said.

Mr. Trump this week proposed “extreme vetting” for those seeking admission to the U.S., saying he wants to make sure those who come here must show fealty to American values of pluralism.

Democrats and immigrant-rights advocates mocked his proposal, saying it sounded un-American.

U.S. law already requires immigrants to show they are “attached” to the values of the Constitution, and throughout the Cold War those who espoused communist beliefs were to be denied. Some analysts said Mr. Trump’s new plan could be seen as an update of the Cold War plans for the modern era of terrorism.

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