Trump and Putin have been on the attack since their historic summit in Helsinki on July 14.
The president has lashed out at the leaders of other nations, including China, that are “reluctant to play ball with the US and European Union,” and called on Russia to pull out of the Paris climate accord.
Putin, however, has remained upbeat about his relationship with the new American president.
Trump has already met Putin in a pair of private meetings, and his relationship has been described as “close.”
Trump’s recent tweets have been especially damaging to Russia’s relations with NATO allies, which have been at the center of a Cold War rivalry between Moscow and the West since the collapse of the Soviet Union.
The Kremlin is trying to portray itself as a guarantor of peace and stability for the region.
“The Trump administration has shown the world that it has no intention of recognizing the sovereignty and territorial integrity of Ukraine,” Putin told the RIA Novosti news agency.
Trump and his team have also accused Russia of trying to interfere in the 2016 presidential election.
“Russia is using every means at its disposal to undermine the sovereignty, territorial integrity, and political independence of the Russian Federation,” he said.
In his latest tweet, Trump also said that Russia has been trying to destabilize Ukraine and that its actions have been “wrong and harmful.”
“Russia has tried to destabilise Ukraine with cyber-attacks, destabilizing the international system and destabilizing our national interests,” Trump said.
Putin has denied the claims.