The witnesses described the story this way: Mr. [pro-golfer Bernard] Langer, a 59-year-old native of Bavaria, Germany — a winner of the Masters twice and of more than 100 events on major professional golf tours around the world — was standing in line at a polling place near his home in Florida on Election Day, the president explained, when an official informed Mr. Langer he would not be able to vote.

Ahead of and behind Mr. Langer were voters who did not look as if they should be allowed to vote, Mr. Trump said, according to the staff members — but they were nonetheless permitted to cast provisional ballots. The president threw out the names of Latin American countries that the voters might have come from.

Mr. Langer, whom he described as a supporter, left feeling frustrated, according to a version of events later contradicted by a White House official.

The anecdote, the aides said, was greeted with silence, and Mr. Trump was prodded to change the subject by Reince Priebus, the White House chief of staff, and Senator John Cornyn, Republican of Texas.

Just one problem: Mr. Langer, who lives in Boca Raton, Fla., is a German citizen with permanent residence status in the United States who is, by law, barred from voting, according to Mr. Langer’s daughter Christina.

“He is a citizen of Germany,” she said, when reached on her father’s cellphone. “He is not a friend of President Trump’s, and I don’t know why he would talk about him.”

She said her father was “very busy” and would not be able to answer any questions.

But a senior White House staff member, who was not at the Monday reception but has heard Mr. Trump tell the story, said Mr. Langer saw Mr. Trump in Florida during the Thanksgiving break and told him the story of a friend of Mr. Langer’s who had been blocked from voting.