President Trump called in a new interview for the Republican Party to "nationalize" voting in the United States, an aggressive rhetorical step that was likely to raise new worries about his administration's efforts to involve itself in election matters.

During an extended monologue about immigration on a podcast released on Monday by Dan Bongino, his former deputy F.B.I. director, Mr. Trump called for Republican officials to "take over" voting procedures in 15 states, though he did not name them.

"The Republicans should say, 'We want to take over,'" he said. "We should take over the voting, the voting in at least many — 15 places. The Republicans ought to nationalize the voting."