Asked whether it was irresponsible for the president to undermine public confidence in the integrity of the vote without data to back up his assertion, Trump said, “It doesn’t have to do with the vote.”

“It has to do with the registration,” he continued. “And when you look at the registration, and you see dead people that have voted, when you see people that are registered in two states — and that voted in two states — when you see other things, when you see illegals, people that are not citizens and they are on the registration rolls. Look, Bill, we can be babies, but you take a look at the registration. You have illegals, you have dead people, you have this — it’s really a bad situation. It’s really bad.”

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O’Reilly pressed: “But the data has to show that 3 million illegals voted” for Trump to be right, he said.

“Forget that,” Trump replied. “Forget all that. Just take a look at the registration, and we’re going to do it. And I’m going to set up a commission to be headed by Vice President Mike Pence, and we’re going to look at it very carefully.”

“Forget all that” is about as close as Trump gets to acknowledging he was wrong to claim that millions of illegal ballots tainted an election in which he lost the popular vote. He is basically asking voters to pretend that what he really meant all along is that registries often contain inaccuracies.

That is not what Trump meant all along, but he is right about the registries. A 2012 Pew Charitable Trusts study that Trump’s team often cites (and mischaracterizes) concluded that “approximately 24 million registration records, or nearly 13 percent of the national total, are estimated to be inaccurate or no longer valid.”

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It is common for people who move from one state to another to be registered in both because of faulty record keeping, although there is no evidence that voting twice is an epidemic. People registered in more than one state during the election included Trump’s younger daughter, Tiffany; senior adviser Stephen K. Bannon; son-in-law and adviser Jared Kushner; and press secretary Sean Spicer.

Trump and O’Reilly, who have been friendly for 30 years, have a somewhat complicated on-camera relationship.

Trump was a frequent guest on O’Reilly's show during the campaign and was generally more tolerant of tough questions from his milkshake buddy than from other interviewers. When Trump grew angry at Fox News’s coverage during the Republican primary and vowed not to appear on the network, he ended his brief boycott on “The O’Reilly Factor.”

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But Trump at times has bristled at O’Reilly’s criticism. In a post-debate interview last March, for instance, Trump suggested that O’Reilly should consult a psychiatrist because he had “become very negative.”

The Super Bowl sit-down, filmed Friday, was Trump’s first interview with O’Reilly since late October, when O’Reilly repeatedly challenged Trump’s claims that the election would be rigged against him.

“On a mass level, you don’t believe that [with] 120 million American votes that the thing is going to be dishonest, do you?” O’Reilly asked. “Do you believe that?”

O’Reilly had told viewers a couple of days earlier that “the vote cannot be possibly rigged . . . and Mr. Trump should accept whatever happens at that vote, unless there is compelling evidence of corruption. Undermining our electoral system is not a patriotic thing.”

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O’Reilly said at an event in Scottsdale, Ariz., on Tuesday that he had spoken to Trump privately since then. Specifically, he said he told the president that political opponents would try to impeach him if he did not act “with a little more subtlety.”