They spoke towards the end of a formal dinner but the White House has not revealed what was discussed.
President 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 Moscow tried to tip the election in Mr Trump's favour, something denied by Russia. Mr Trump has rejected allegations of any collusion.
The extra conversation happened during a private meal of heads of state at the G20 summit in Hamburg earlier in the month.
The Kremlin said at the time that the two leaders had had "an opportunity to continue their discussion during the dinner" but the extent of the meeting was not known.
Mr Trump left his seat and headed to Mr Putin, who had been sitting next to Mr Trump's wife, Melania, US media said. The US president was alone with Mr Putin, apart from the attendance of the Russian president's official interpreter.
Mr 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.
Mr 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.
Mr Bremmer 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 characterised as a meeting."
Mr 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, Mr Trump said he had repeatedly pressed Mr 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 Tuesday, the Senate intelligence committee said it wanted to interview Mr Trump's son, Donald Jr, and other members of the Trump team, over a meeting they had with a Russian lawyer in June last year.
Mr Trump Jr said he had attended the meeting with Natalia Veselnitskaya as he was promised damaging material on Hillary Clinton, but it did not materialise.
On Wednesday, Ms Veselnitskaya told Russia's RT television channel she would be willing to testify before the Senate on the matter.
Meanwhile, the White House said Mr 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.
Mr 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.