Obama was sworn in as the 44th president of the U.S. on Tuesday in Washington, becoming the country's first African-American head of state.
According to the source quoted by Kommersant, Obama will set off for his first European tour in April and is to visit London on April 2 to participate in the G20 summit, and meet for the first time with his Russian counterpart Dmitry Medvedev.
Obama will then attend a NATO summit on April 3-4 in Strasbourg, France and Kehl, Germany to mark the 60th anniversary of the military alliance.
The source said Obama could fly to Moscow after the summit. The previous U.S. president, George Bush, flew to Russia's Black Sea resort of Sochi after last year's NATO summit in Bucharest.
He added however that U.S. Secretary of State Hilary Clinton was likely to visit Russia in March before Obama's trip to Moscow. The Russian Foreign Ministry said the date of Clinton's visit had not yet been agreed on.
Meanwhile, a source in the Russian presidential administration said that no official information on Obama's possible visit to Moscow had been received.
President Medvedev earlier said he hoped the United States would develop constructive relations with Moscow under Obama.
Russia-U.S. relations have been frayed in recent years by Washington's plans to deploy elements of a missile shield to Central Europe, Russia's five-day war with Georgia over South Ossetia last August, and NATO's eastward expansion.
"We want to cooperate with them where we can, and there are a whole host of areas particularly around nonproliferation of weapons and terrorism where we can cooperate, but we also have to send a clear message that they have to act in ways that are not bullying their neighbors," Obama told NBC's meet the press in December, a little more than a month after his election victory.
"They're increasingly assertive. And when it comes to Georgia and their threats against their neighboring countries, I think they have been acting in a way that is contrary to international norms," he added.
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Infographics: World War I, 1914-1918
If attempts to drag Russia into a direct military conflict in Ukraine are successful, it would be a catastrophe for Russia comparable to the 1979-1989 Afghan war. There is no direct evidence that the US is trying to bring about a second Afghan war, but indirect evidence abounds.