The United States has held talks with Russia, and separately with Ukraine and European leaders, on proposals for ending the conflict but no deal has yet been reached despite Trump's repeated promises to clinch one.
Ukraine's European allies, currently arguing in public about Trump's threats against Greenland, are concerned the United States could demand Ukrainian accept territorial concessions.
"Dialogue is constructive and more and more people understand the fairness of Russian position," Putin envoy Kirill Dmitriev said after talks with US envoy Steve Witkoff and Trump's son in law Jared Kushner in the "USA House" at Davos.
Witkoff said: "We had a very positive meeting," Russia's RIA news agency reported.
The meeting lasted for two hours, a source who spoke on condition of anonymity said.
At stake is how to end the deadliest war in Europe since World War II, the future of Ukraine, the extent to which European powers are sidelined and whether or not a peace deal brokered by the United States will endure.
Russia invaded Ukraine in February 2022 after eight years of fighting in eastern Ukraine.
Russia controls about 19 per cent of Ukraine, including the Crimea peninsula which it annexed in 2014, as well as most of the eastern Donbas region, much of the Kherson and Zaporizhzhia regions as well as slivers of four other regions.
The Kremlin says Crimea, Donbas, Kherson and Zaporizhzhia are now parts of Russia.
Ukraine says it will never accept that, and almost all countries consider the regions to be part of Ukraine.
Ukraine and many European leaders say that Russia cannot be allowed to achieve its aims after what they cast as an imperial-style land grab.
If Russia wins, European powers say, then it will one day attack the NATO military alliance.
Russia says such claims are ridiculous and that it has no intention of attacking a NATO member.
Russia says that European leaders are intent on scuttling the peace talks by introducing conditions that they know will be unacceptable to Russia, which took 12 to 17 square km of Ukrainian territory per day in 2025.
Putin casts the war as a watershed moment in relations with the US and its European allies, which he says humiliated Russia after the Soviet Union fell in 1991 by enlarging NATO and encroaching on what he considers Russia's sphere of influence.
He has repeatedly said he is open to peace but one based on the realities of the battlefield.
The United States says a total of a million Russian and Ukrainian men have been killed or injured in the war.
Russia and Ukraine do not publish losses.
