The president and his Russian counterpart spoke towards the end of a formal dinner but the White House has not revealed what was discussed.
Donald Trump has condemned media revelations of the talks as “sick”.
The two leaders’ relationship is under scrutiny amid allegations of Russian interference in the US election.
US intelligence agencies believe Russia tried to tip the election in Donald Trump’s favor, something denied by the Kremlin. Donald Trump has rejected allegations of any collusion.
The extra conversation happened during a private meal of heads of state at the G20 summit earlier in the month.
President Trump left his seat and headed to Vladimir Putin, who had been sitting next to Melania Trump, media said. The president was alone with Vladimir Putin, apart from the attendance of the Russian president’s official interpreter.
Donald Trump had been seated next to Japanese PM Shinzo Abe’s wife, so the US interpreter at the dinner spoke Japanese, not Russian. No media were in attendance.
The length of the talks has been disputed.
Ian Bremmer, president of the US-based Eurasia Group, who first reported them in a newsletter to clients, said: “Donald Trump got up from the table and sat down with Putin for about an hour. It was very animated and very friendly.”
No-one else was nearby, so the topics of discussion were not known, he said.
Ian Bremmer had not been at the dinner but said details were given to him by unnamed attendees who, he said, were “flummoxed, confused and startled” by the turn of events.
He told Bloomberg he had never before seen “two major countries with a constellation of national interests that are as dissident while the two leaders seem to be doing everything possible to make nice-nice and be close to each other”.
In a statement, a senior White House official said there was no “second meeting”, just a brief conversation after dinner.
The official said: “The insinuation that the White House has tried to <hide> a second meeting is false, malicious and absurd. It is not merely perfectly normal, it is part of a president’s duties, to interact with world leaders.”
National Security Council spokesman Michael Anton said it was not a meeting but a “pull aside”, adding: “A conversation over dessert should not be characterized as a meeting.”
President Trump later said on Twitter: “Fake News story of secret dinner with Putin is <sick>. All G20 leaders, and spouses, were invited by the Chancellor of Germany. Press knew!”
The dinner and its attendees have always been known. Only the Trump-Putin discussion had not been reported before.
At the earlier, formal meeting, their first face-to-face encounter, President Trump said he had repeatedly pressed President Putin about the allegations of interference in the US vote.
“I said, <Did you do it?> He said, <No, I did not, absolutely not>. I then asked him a second time, in a totally different way. He said, <Absolutely not>.”
There are congressional investigations, and one by a special counsel, into the allegations of Russian interference in the US election and possible collusion with the Trump team.
On July 18, the Senate intelligence committee said it wanted to interview Donald Trump’s son, Donald Jr., and other members of the Trump team, over a meeting they had with a Russian lawyer in June 2016.
Donald Trump Jr. said he had attended the meeting with Natalia Veselnitskaya as he was promised damaging material on Hillary Clinton, but it did not materialize.
On July 19, Natalia Veselnitskaya told Russia’s RT TV channel she would be willing to testify before the Senate on the matter.
Meanwhile, the White House said President Trump would nominate former Utah governor Jon Huntsman as ambassador to Russia, a key post for a president who promised to improve relations with Moscow.
Jon Huntsman, who served as ambassador to China and Singapore, needs to have his name confirmed by the Senate.
The suspicions over Russian interference are likely to play a significant factor in his confirmation process, correspondents say.
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