As he spoke, news was breaking out of Washington that Putin had been invited to the White House this fall.
"This is the beginning of the dialogue with Russian Federation and our administration and theirs and we're going to continue working through those things", she told reporters at the White House on Wednesday.
He also reiterated denials of Russian involvement in the poisoning of former Russian spy Sergei Skripal in Britain.
Mr Trump also accused the news media of trying to provoke a confrontation with Moscow that could lead to war, although concerns about the meeting have been raised by a broad cross-section of Republicans and Democrats.
Trump tweeted that the two men discussed Ukraine, but has not mentioned a referendum or revealed specifics.
"I believe that it has to become normal for a US President and Russian [President] to meet", adding that it is always good "especially when these two countries talk" because they control 90% of the world's nuclear arsenal.
In an interview with "CBS Evening News" anchor Jeff Glor Wednesday, President Trump was asked if he holds Putin personally responsible for meddling. We are doing MUCH better than any other country!'
Mr Trump tweeted: "The Fake News Media wants so badly to see a major confrontation with Russian Federation, even a confrontation that could lead to war".
Meanwhile, Homeland Security Secretary Kirstjen Nielsen said she still has not seen evidence that Moscow tried to help elect Trump. "I don't think he was prepared as well as he should have been".
In a possible dig at Trump's unpredictable presidency, Putin vaunted Russia's "consistent, responsible, independent foreign policy".
The Kremlin has the final say, but hasn't responded yet to Trump's invitation.
In his first public comments about the summit, Mr Putin told Russian diplomats that US-Russian relations are "in some ways worse than during the Cold War".
Still, Trump has expressed a preference for the White House setting for major meetings, including floating an invitation to Washington for North Korea's Kim Jong-un after their meeting in Singapore last month.
"So just to repeat it, I said the word "would" instead of "wouldn't" and the sentence should have been, and I thought I would maybe be a little bit unclear on the transcript or unclear on the actual video, but the sentence should have been, 'I don't see any reason why it wouldn't be Russian Federation".
The day after that, asked whether Russian Federation posed a threat to the United States, Trump said "no" - though hours later he told CBS News in an interview he gave a stern warning to Putin not to interfere in this year's congressional elections, a different take on the meeting than he provided previously.
This is not the first time the magazine has depicted Trump on the cover. Nearly nine in 10 Democrats said they believed USA intelligence agencies that Russian Federation interfered in the 2016 election, while only about half of Republicans accepted that conclusion.
House Democratic leader Nancy Pelosi urged House Speaker Paul Ryan to make clear that Putin wouldn't be invited to address Congress if he visits Washington.
A White House official who spoke on condition of anonymity did not specify if that included Russia's interference in U.S. elections.
What else has happened since the Trump-Putin summit?
In response to Trump's tweet, Clinton posted a more recent video, sharing her remarks from a 2016 campaign rally.
The financier drew Putin's ire after he successfully advocated in 2012 for the economic sanctions. "Somehow America seems to forget the history of Russia's efforts to undermine western democracy for decades now", Pompeo said. He cited USA sanctions and the expulsion of alleged Russian spies from the US, telling reporters that Putin "understands it, and he's not happy about it". Both are prominent Trump critics.
Sanders called the legislation "hypothetical" and declined to say whether the president would back it.
The resolution was one of several brought by lawmakers, including one brought by Sen.
The Senate unanimously passed a resolution Thursday expressing opposition to the notion the United States might cooperate with the Russian government to make former USA diplomats, officials, and military members available for questioning. A similar House vote Tuesday failed on a party-line vote.
As for whether Trump agreed with national intelligence director Coats that the Russian threat continues, Trump said he accepts that.