“Fantastic job — what a good job he did,” Mr. Trump said, adding wryly, “Tomorrow he will be announcing that he’s running for office.”

The episode was the latest example of the president making a racially tinged remark in public, this time during an event devised to spotlight his tough immigration agenda. It came only days after a former aide, Omarosa Manigault Newman, claimed in a tell-all book that Mr. Trump had been caught on tape using a racial slur to refer to African-Americans, a charge that he has denied but that his press secretary did not flatly rule out.

Mr. Trump kicked off his presidential campaign by denouncing Mexican immigrants as criminals and rapists, and, as he did at Monday’s gathering, he routinely raises the specter of the brutal transnational gang MS-13, whose members he calls “animals,” when speaking about the need for tighter immigration laws.

Hector Garza, a colleague of Mr. Anzaldua’s who is also the vice president of the National Border Patrol Council, which endorsed Mr. Trump during the 2016 election, said he was proud of his friend and did not see anything wrong with the president’s remark about his language skills. Mr. Garza said Mr. Anzaldua had a stellar record — including the “Top Dog” award for narcotics apprehensions.

“I didn’t think anything of it,” said Mr. Garza, who said he had met with Mr. Trump numerous times and who, like Mr. Anzaldua, is Hispanic. “What ends up happening is that President Trump feels very comfortable with our agents, and our agents support President Trump, and I think this was him speaking comfortably with one of our agents. I didn’t see anything wrong with it.”