Bolton did not detail the Russian president's reaction, though earlier in the day a Kremlin spokesman made clear the country was against the move.
Putin and Bolton laughed as they exchanged the quips at the start of the meeting, setting an easy tone ahead of what was expected to be a tense discussion given Trump's weekend statement about abandoning the Intermediate-Range Nuclear Forces (INF) Treaty.
The INF Treaty was signed by Mikhail Gorbachev and Ronald Regan, according to which, all USA and Soviet-Union land-based ballistic and cruise missiles, ranging between 500 and 5,500 kilometers must be banned.
Earlier Tuesday, Putin's spokesman Dmitry Peskov said Trump took "a risky position" by deciding to abandon an existing nuclear weapons treaty with Russian Federation without offering anything to replace it.
Bolton's Kremlin meeting with Putin followed two days of talks with top Russian foreign policy and security officials. "It is Russia's position that they are not in violation", Bolton told reporters, noting that officials in the Obama administration had also alleged Russian violations of the pact dating back at least to 2013 - and had also been met with denials.
"The point I made to Russian colleagues today was that I didn't think, whatever they had done in terms of meddling in the 2016 election, that they had any effect on it", Bolton told reporters Monday, "but what they have had an effect in the United States is to sow enormous distrust of Russia".
U.S. National Security Adviser John Bolton (left) and Russian Security Council chairman Nikolai Patrushev talk prior their official talks in Moscow on October 22, 2018.
"Do they really not understand in Washington what this could lead to?" "Not only Russian Federation, but all those who cherish the world, especially a world without nuclear weapons, must declare this".
"And it's a major obstacle to achieving agreement on issues where our national interest may converge, so I said, just from a very cold blooded cost benefit ratio, that you shouldn't meddle in our elections because you're not advancing Russian interest, and I hope that was persuasive to them", he added.
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US officials going back to the Obama administration have accused Russian Federation of deliberately deploying a land-based cruise missile to pose a threat to North Atlantic Treaty Organisation.
China was not a party to the original agreement, and Trump argued that it should be included in the treaty. He added that the US would trigger the six-month withdrawal mechanism from the INF treaty "in due course".
Apart from the INF treaty controversy, Bolton said that United States and Russian Federation would resume dialogue on counter-narcotics and human trafficking, as well as on counter-terrorism.
Trump said, "We'll have to develop those weapons, unless Russian Federation comes to us and China comes to us and they all come to us and say let's really get smart and let's none of us develop those weapons, but if Russia's doing it and if China's doing it, and we're adhering to the agreement, that's unacceptable".
Beatrice Fihn, the executive director of the International Campaign to Abolish Nuclear Weapons - the coalition that won the 2017 Nobel Peace Prize - said, "By declaring he will leave the INF Treaty, President Trump has shown himself to be a demolition man who has no ability to build real security".
On Wednesday, Peskov said Russian and the US had also discussed the possibility of Putin visiting Washington, D.C.
China is not part of the INF agreement.
Despite threatening to spend vast sums to increase the United States nuclear arsenal, Trump said ultimately he hoped to bring the country back onto the path of reducing its weapons stockpiles. "They think if they threaten the West properly, it gets scared".