"Both President Trump and President Putin think they may be able to find constructive solutions," Bolton said at a news conference in Moscow after his day of meetings. "I'd like to hear someone say that's a bad idea."
The summit is expected to take place in mid-July somewhere outside Russia, during Trump's trip to Europe for a previously scheduled Nato summit meeting on July 11 and 12 in Brussels and a visit to Britain on July 13.
There is speculation that Trump and Putin could meet in Helsinki or Vienna, but neither US nor Russian officials have confirmed the location.
Looking ahead to his Putin meeting, Trump told reporters, "I've said it from day one, getting along with Russia and China and with everybody is a very good thing."
The President praised the Russians for doing "a fantastic job" hosting the World Cup, complimenting the quality of the venues and saying the matches have been "exciting even if you are not a soccer fan."
Thomas Wright, director of the Centre on the United States and Europe at the Brookings Institution, said the summit announcement "is a sign of Trump unbound."
"He wants to work with Putin. This is what he's told people he's going to do and he's not listening to any objections," Wright said, noting that Trump "was always a reluctant participant" in implementing sanctions and other tough measures against Russia.
Trump's summit with Putin threatens to further rupture his relationship with European leaders and is likely to raise additional doubts about his commitment to America's traditional alliances.
It will also garner scrutiny in light of Special Counsel Robert Mueller's ongoing investigation into Russia's election interference and possible collusion between Putin's government and Trump's campaign.
Bolton dismissed those concerns while chiding Trump's domestic detractors.
"A lot of people have said or implied over time that a meeting between President Trump and President Putin would somehow prove some nexus between the Trump campaign and the Kremlin, which is complete nonsense," Bolton said.
Trump, he continued, believes that "what must guide his conduct of American foreign policy is the pursuit of American national interests, and he judges, correctly in my view, that this bilateral summit between himself and President Putin is something that he needs to do and will do regardless of political criticism at home."
In Washington, Trump's critics worry that he might shrink from directly confronting Putin on the substantial and serious differences between their countries.
They cited not only Russia's election interference, but also its alleged poisoning of a former Russian double agent in Britain, its support for Syrian President Bashar al-Assad's murderous government and its occupation of Crimea in Ukraine.
Senator Christopher Coons, (D), a member of the Senate Foreign Relations Committee, said in an interview that a Trump-Putin summit could be constructive and noted that US presidents regularly met with Russian leaders at the height of the Cold War.
But, Coons said, "I'm very concerned that President Trump can't help himself but try to please another autocrat at the expense of our democracy."
"If we can have a better and stronger relationship with any country in the world and can directly address challenges we have with them, that's to our benefit," Coons added.
"That's the point of diplomacy over military conflict. But it doesn't advance America's interest to knuckle under or brush aside significant threats to our security or our allies' security."
At the Kremlin, Putin warmly greeted Bolton in a grand oval meeting hall, flanked by statues of Russian tsars set before lime-painted walls.
The Russian President opened the meeting by repeating to Bolton his frequent contention that US-Russian relations are in a poor state in large part because of the domestic political environment in the United States.
"Your visit here to Moscow inspires hope that we will be able to take first steps to restore full-fledged relations between Russia and the United States," Putin said.
"Russia never sought confrontation, and I hope that today we will be able to talk about what we can do from both sides in order to restore full-fledged relations on the foundation of equality and of respect for each other's interests."
Bolton's engagement with Putin on Wednesday contrasted sharply with the longtime hawk's harsh criticism of the Russian President before he took over as Trump's National Security Adviser in April. Last year, Bolton described Russian interference in the 2016 election as "a true act of war" and concluded: "Negotiate with Russia at your peril."
