The radio program remarks were reported by the website BuzzFeed.

On Tuesday afternoon, Mr. Trump, in an email sent by a spokesman, issued his most forceful rebuke of Mr. Duke to date: “As we have many times in the past, we disavow David Duke. There’s no place for this in the Republican Party.”

Questions about Mr. Duke have dogged Mr. Trump for months, after he was slow to disavow him in a CNN interview during the South Carolina primary when Mr. Duke spoke admiringly of the candidate’s message. Mr. Trump later said, “I disavow,” and he criticized Mr. Duke’s suggestion that he had conquered “Jewish supremacists” within the Republican Party.

In his own campaign announcement in recent days, Mr. Duke said that Mr. Trump was championing issues that he had raised for decades.

When Mr. Trump was asked Sunday on NBC’s “Meet the Press” if he would support a Democrat over Mr. Duke in the general election, he replied: “I guess depending on who the Democrat is, but the answer would be yes. Look, the answer is as quick as you can say it — in fact, I want to answer you before you — because last time with another person in your position, I did it very quickly and they said he didn’t do it fast enough. Rebuked.”