"It's coming from all over South and Latin America, and it's coming probably from the Middle East," he added. "We have no protection and we have no competence, we don't know what's happening. And it's got to stop and it's got to stop fast."
To make it stop, he proposed building a wall along the U.S.-Mexico border and insisted he could make the Mexican government pay for it. He has also asserted that English, rather than Spanish, should be spoken in the U.S.
What Republican leaders fear, after a 2012 election in which President Barack Obama received 7 in 10 Latino votes, is that their presidential prospects will pay the price through further alienation.
Those fears have increased as Trump has continued to hold the lead in polls.
Business-oriented Republicans have counted on Trump's rivals to outpace him with a more temperate message. The leading fund-raiser in the GOP race, former Florida Gov. Jeb Bush, relies on a compelling personal story — his marriage to a Mexican-American he met while studying abroad.
"In 1971, eight years before then-candidate Ronald Reagan said that we should stop thinking of our neighbors as foreigners, I was ahead of my time in cross-border outreach," Bush said in announcing his candidacy. "Across a plaza, I saw a girl. She spoke only a little English. My Spanish was OK but not that great. With some intensive study, we got that barrier out of the way in a hurry.
"It has been a gracious walk through the years with the former Columba Garnica de Gallo," Bush concluded. "As a candidate, I intend to let everyone hear my message, including the many who can express their love of country in a different language."
Pressure from conservatives has forced Bush to temper his views on immigration just the same. He once favored a "path to citizenship" for the 11 million undocumented immigrants in the U.S. Now, he favors only a path to "legal status" instead.