Despite those numbers and the skepticism from colleagues, Mr. Kushner has been advising Mr. Trump that black voters can be converted into supporters if they are simply educated on his policies. Mr. Trump’s biggest challenge, Mr. Kushner has told people, is a “knowledge gap” on many of the president’s accomplishments, particularly on the issue of criminal justice reform, which Mr. Kushner has spearheaded.

Mr. Kushner has hosted black leadership summits at the White House. And two of the lone African-American White House staff members, Scott Turner and Ja’Ron Smith, have been traveling to black communities to make a pitch for Mr. Trump.

Even some of Mr. Trump’s most vociferous Republican critics said it was not an off-the-wall idea, although Mr. Trump had only a thin record to run on and a political career marked by racial invective and a proposed Muslim ban to run from.

“They don’t need to reinforce the base at this point,” said William Kristol, the conservative writer and prominent “Never Trump” Republican. “If they can chip away a little bit at the African-American vote, or make it harder to mobilize them against Trump, it’s not a foolish expenditure of money.”

Mr. Kushner has pointed people to public polls, including Rasmussen, that are outliers but that have shown Mr. Trump winning up to 30 percent of the African-American vote. In 2016, Mr. Trump won just 8 percent of the black vote, according to exit polls.

But Mr. Kushner has told people that in Hillary Clinton, the 2016 Democratic nominee, they faced a candidate who had been popular in the black community for decades, while Mr. Trump was then an unknown quantity. In 2020, Mr. Kushner has argued, voters need to be reminded of the president’s accomplishments and that their “worst fears have not been realized.”

Mr. Trump’s goal is finding a way to shave off enough supporters from the Democratic nominee to compensate for an expected loss of suburban voters in swing states like Michigan and Pennsylvania.