The White House has confirmed the discussion came later, after Mr Putin and Mr Trump held formal talks for more than two hours.
It was the most highly anticipated meeting of the G20 summit in Germany earlier this month.
The United States and Russian presidents were meeting for the first time face-to-face.
It came amid investigations in the United States into possible collusion between Russia and US president Donald Trump's campaign people during last year's presidential election.
The meeting lasted more than two hours, despite being scheduled for only 30 minutes.
Afterwards, Russia's foreign minister claimed Mr Trump accepted Vladimir Putin's assurances Russia did not meddle in the election, a claim the White House later denied.
Now, it has been revealed the leaders later spoke for a second time, a previously undisclosed discussion.
But White House deputy press secretary Sarah Huckabee Sanders is defending those talks, saying they were no secret.
"This was a publicly disclosed event. The President participated in an official dinner of the G20. That was part of his schedule. That was released publicly. You guys came and took pictures of it. It wasn't like this was some sort of hidden dinner. The pictures have been replayed over and over. It was part of the official G20 schedule. So to act as if this was some secret is just absolutely absurd. They had a brief conversation, and I'm not going to get into the specifics of the conversation, but, again, this was a social dinner, where the President spoke with many world leaders, as is the purpose. I think it would be incredibly awkward for them to all sit at a dinner and not speak to each other, and I would imagine that all of you would agree with that."
But the president of the Eurasia Group, Ian Bremmer, who first revealed the talk, says onlookers have described that so-called "brief conversation" as nearly an hour long.
"About halfway into the dinner, Trump gets up, he walks around the table, goes and sits down with Putin. Only Putin's translator, his interpreter, is there. And they have a marvellous time chatting closely, very engaged, for about an hour. And this was noted by a lot of people that were around the room. They found it unusual, surprising."
Mr Bremmer was told Mr Trump was very animated as he spoke with Mr Putin, often using his hands to gesture.
The topic of conversation is still unknown.
"It's not that the idea is the American president can't have a pull-aside with the Russian president in a G20 meeting. Nothing wrong with that. That happens all the time. When you take a step back and put it in the broader context of the US-Russia relationship and Putin and Trump, and where the world is heading, you realise there's something here that does not add up."
Mr Trump has defended the dinner in a pair of angry tweets that noted the dinner had been on his public schedule.
German chancellor Angela Merkel's spokesman, Steffen Seibert, has also defended the conversation.
"It is the fundamental point of the G20 meeting that, alongside the working meetings, there's room and opportunity for multiple informal contacts, talks and meetings, and that is certainly the point of such a dinner. This dinner at the Elbphilharmonie did, exactly as hoped, produce a lively exchange between many and many others, particularly at the end when the seating arrangements dissolved somewhat. And that is what I can tell you about this. Spokespeople for the heads of state and government involved will have to tell you about conversations that were conducted here. The Chancellor, in any case, was not surprised nor confused by anything."
The news comes as Russian lawyer Natalia Veselnitskaya, who met with Mr Trump's oldest son during the election campaign, says she is ready to testify before the US Senate.
In emails publicly released last week, Mr Trump Junior agreed to meet with her after receiving an email offering him very high-level and sensitive information.
That information would allegedly incriminate Democratic presidential nominee Hillary Clinton, but Ms Veselnitskaya says that is not the way it was.
"If the Senate wants to hear the real, true story, I, with great readiness, will tell them about everything I wanted to tell Mr Trump (Junior) and what I wanted to say in Congress last year but the program was changed -- absolutely everything I know about, in what ways huge millions of money came into my country Russia and returned as billions and there were no taxes paid. But who they went to, that is not a question to me, but please refer to the United States system."
Donald Trump Junior and former campaign chairman Paul Manafort have been scheduled to testify before the Senate judiciary committee next week.
Donald Trump's son-in-law, Jared Kushner, is also expected to appear before the Senate intelligence committee but behind closed doors.
