In the earlier, formal meeting, Mr. Trump said later, he asked the Russian president twice about his role in the American vote. Mr. Putin denied involvement, and the two men agreed to move beyond the dispute in the interest of finding common ground on other matters, including a limited cease-fire in Syria.

There is no official United States government record of the intimate dinner conversation, because no American official other than the president was involved.

“Pretty much everyone at the dinner thought this was really weird, that here is the president of the United States, who clearly wants to display that he has a better relationship personally with President Putin than any of us, or simply doesn’t care,” said Ian Bremmer, president of the Eurasia Group, a New York-based research and consulting firm, who said he had heard directly from attendees. “They were flummoxed, they were confused and they were startled.”

The encounter occurred more than midway through the lengthy dinner, when Mr. Trump left his chair and approached Mr. Putin, who had been seated next to the first lady, Melania Trump. It was described to Mr. Bremmer by other guests as lasting roughly an hour and not initially disputed by a White House official who spoke on the condition of anonymity.

But Sean Spicer, the White House press secretary, disputed that account. He said Mr. Trump had described the exchange with Mr. Putin as purely social, and as lasting far less than an hour. “It was pleasantries and small talk,” Mr. Spicer said. In a separate statement, the White House said the two presidents had spoken through the Kremlin’s interpreter because the American translator with Mr. Trump did not speak Russian.

Experts in United States-Russia relations said such an encounter — even on an informal basis at a social event — was a concern because of its length, which suggested a substantive exchange, and because there was no note taker or national security or foreign policy aide present.

“We’re all going to be wondering what was said, and that’s where it’s unfortunate that there was no U.S. interpreter, because there is no independent American account of what happened,” said Steven Pifer, a former ambassador to Ukraine who also specializes in Russia and nuclear arms control.