Mr. Trump’s effort to ease tensions came in tandem with a visit to Russia by Chancellor Angela Merkel of Germany, who met with Mr. Putin in the southern resort city of Sochi.
But the issues dividing Russia and the West remain unresolved, and the two sides appear no closer to agreement over the Syrian civil war, the Russian intervention in Ukraine or other disputes.
Further complicating the relationship are the continuing investigations into Russian meddling in last year’s presidential elections in the United States, which American intelligence agencies have said was ordered by Mr. Putin to help tilt the vote to Mr. Trump. European leaders have pointed to signs of Russian interference in their own elections lately.
At a news conference with Ms. Merkel on Tuesday, Mr. Putin dismissed allegations that Russia was seeking to influence the political landscape in the West by supporting far-right parties and undercutting mainstream factions. “We never interfere in the political life and the political processes of other countries, and we don’t want anybody interfering in our political life and foreign policy processes,” Mr. Putin said.
Mr. Putin has stuck by Mr. Assad even as much of the rest of the world has called for him to step down after six years of grinding civil war that has left more than 400,000 dead. Russia vetoed a United Nations Security Council resolution condemning the chemical weapons attack.
Asked about Syria at the news conference with the German chancellor, Mr. Putin said that the two sides had discussed settling the conflict there. He emphasized that cooperation with Washington was critical.
“Certainly, without the participation of such a party as the United States, it is also impossible to solve these problems effectively,” Mr. Putin said. “So we are and will continue to be in contact with our American partners, and I hope that we will attain understanding there regarding joint steps in this very important and sensitive area of international relations today.”
When he was asked whether he had the influence to sway Mr. Assad, Mr. Putin said that Russia, in tandem with Turkey and Iran, was trying to “create the conditions for political cooperation from all sides.”
A cease-fire is the main priority, Mr. Putin said. It will be the focus of talks involving various parties to the conflict that are to take place on Wednesday and the next day in Astana, the capital of Kazakhstan. The United States thus far has not had any important role in those talks, which Russia, Iran and Turkey set up outside the previous system of negotiations in Geneva.
Mr. Trump and Mr. Putin have spoken by telephone twice since the American inauguration in January, the first time about a week after the president was sworn in and the second in early April, when Mr. Trump called to express condolences over a terrorist attack in St. Petersburg, Russia.
The two men may have their first meeting in person as presidents on the sidelines of a Group of 20 summit meeting in Germany in July.