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.
Analysis: Uncomfortable questions raised
Jonathan Marcus, BBC News defence and diplomatic correspondent
Given the poor state of relations between Washington and Moscow and the controversy surrounding Russia's efforts to interfere with the US presidential campaign, each and every encounter between Mr Putin and Mr Trump is bound to be carefully scrutinised.
Thus the apparently impromptu discussion between the two men at the G20 dinner inevitably raises many questions. What was President Trump seeking to do in approaching the Russian president? Were matters of substance discussed? If so, why was no formal note taken? And why did the US president have to rely upon a Russian official for translation?
This is all highly unusual, especially at a time when relations between the two countries are laden with so many problems.
Mr Trump also appeared unaware of another dimension - the message that his tete-a-tete would send to other leaders in the room, who must have watched the US president's gambit with some unease.
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.
"At summit meetings you have little 'pull-asides' between heads of state to discuss business all the time - a one-hour pull-aside is highly unusual in any context," he told the BBC.
"A one-hour pull-aside between Putin and Trump where only the Kremlin translator is there, where we don't know what's discussed, given the uniqueness of the US-Russia relationship... makes the [US] president, surprisingly and disturbingly, not credible."
In a statement, a senior White House official said there was no "second meeting", just a brief conversation after dinner.
The Fake News is becoming more and more dishonest! Even a dinner arranged for top 20 leaders in Germany is made to look sinister!
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: "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.