WASHINGTON — On Monday, President Trump gathered House and Senate leaders in the State Dining Room for a get-to-know-you reception, served them tiny meatballs and pigs-in-a-blanket, and quickly launched into a story meant to illustrate what he believes to be rampant, unchecked voter fraud.

Mr. Trump kicked off the meeting, participants said, by retelling his debunked claim that he would have won the popular vote if not for the three million to five million ballots cast by “illegals.” He followed it up with a Twitter post early Wednesday calling for a major investigation into voter fraud.

When one of the Democrats protested, Mr. Trump said he was told a story by “the very famous golfer, Bernhard Langer,” whom he described as a friend, according to three staff members who were in the room for the meeting.

In the emerging Trump era, the story was a memorable example, for the legislators and the country, of how an off-the-cuff yarn — unverifiable and of confusing origin — became a prime policy mover for a president whose fact-gathering owes more to the oral tradition than the written word.

The three witnesses recall the story this way: Mr. 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.

Either way, the tale left its mark on Mr. Trump, who is known to act on anecdote, and on Wednesday redoubled his efforts to build a border wall and crack down on immigrants crossing the border from Mexico.

The story, the aide added, had made a big impression on Mr. Trump.