Immigration has become a “threshold issue” that motivates voters almost as a proxy for how they feel about the president, said Sam Nunberg, a former aide to Mr. Trump’s presidential campaign who has been studying how to galvanize base voters with immigration hawks in the party like Stephen K. Bannon, the former White House chief strategist.

Explaining how Trump supporters were likely to view illegal immigration, Mr. Nunberg said, “Once again, we believe that Donald Trump is under attack, and that the Democrats are orchestrating it.”

Across the country — from states with small foreign-born populations like Montana to ones where immigrants are a major political constituency like Florida — Republicans have borrowed from Mr. Trump’s playbook. In Indiana they are attacking Senator Joe Donnelly, a vulnerable Democrat, as “Mexico Joe,” a reference to a questionable accusation that he profited after his brother’s business outsourced jobs. In Missouri, Josh Hawley has repeatedly accused his opponent, Senator Claire McCaskill, of favoring “open borders,” a misleading term the president often deploys against Democrats to falsely assert that they oppose enhancing border security.