Trump, whose victory over Hillary Clinton surprised virtually everyone, has given Europeans ample cause for concern.
The irony is cruel: In the name of a peaceful transition, the American president, having thoroughly denounced the billionaire Republican during the recent campaign, must now do his best to reassure his European counterparts about the future of American democracy under a President Trump.
"I think the design of the trip was meant to just give everybody some reassurance that we made it through this campaign and we're going to come out of it all right," said Heather Conley of the Center for Strategic and International Studies, in Washington. "We just have a different scenario now."
The bombastic populist, whose victory over Hillary Clinton surprised virtually everyone, has given Europeans ample cause for concern.
He has questioned the relevance of some of America's paramount alliances, starting with NATO; put the Paris climate-change accord in doubt by calling global warming "a hoax," and sharply criticized the strenuously negotiated pact that Washington and five other countries signed with Iran to curb its nuclear program.
Trump's attitude toward Russian President Vladimir Putin -- the New York mogul called him "a leader, far more than our president has been a leader" -- is deeply concerning in Europe, particularly in small countries like the Baltic nations living in Russia's shadow.
Beyond the many concerns over the future of American foreign policy, many European Union countries are bracing for a possible ripple effect of the outspoken Republican's victory.
"They are very worried, because the same populist, nationalist expressions" that Trump exposed in America on immigration and trade could amplify the already "very strong political currents within Europe," Conley said.
She noted that several European countries have important elections coming up, not least of them the French presidential election next spring.
In Greece on Tuesday for his first visit there, Obama is set to meet with President Prokopis Pavlopoulos and Prime Minister Alexis Tsipras. He is expected to thank them for the "remarkable generosity" the Greek government and people have shown to the thousands of immigrants who have poured into the country from Africa and the Middle East.
During a day heavy with symbolism, Obama on Wednesday will visit the Parthenon in Athens, then deliver a speech -- sure to have considerable resonance, given the recent US elections -- on the challenges of globalization.
His advisers, pointing to the results of that election but also to the equally stunning British vote to leave the European Union, said the US president would offer his thoughts on the reasons so many people in the world feel "like decisions are made beyond their control."
Speaking at the United Nations in September, at a time when the US presidential campaign was in full stride but a Trump victory seemed anything but certain, Obama had called on his fellow leaders to come to grips with the rising frustrations fueling populist movements. He warned them against succumbing to a "soulless capitalism that benefits only the few."
"Twenty-five years after the Cold War, the world is less violent and more prosperous," he said, "and yet there is uncertainty and strife."
"This is the paradox that defines our world today," he said, stressing that a world in which one percent of the people control as much wealth as the other 99 percent can never be stable.
For his sixth visit to Germany since coming to power in 2009, the Democratic president will again meet with Chancellor Angela Merkel, long one of his closest foreign partners, according to Ben Rhodes, the US national security adviser.
The day after his election, the chancellor pointedly reminded Trump of the criteria that have long bound the two countries in close cooperation: "democracy, freedom, as well as respect for the rule of law and the dignity of each and every person, regardless of their origin, skin color, creed, gender, sexual orientation or political views."
Obama will also meet in Germany with French President Francois Hollande -- who once said Trump's "excesses" made people "want to retch" -- British Prime Minister Theresa May, and Italian Prime Minister Matteo Renzi. The leaders plan to discuss the crises in Syria and Ukraine, as well as the fight against the Islamic State group.
The American president will conclude his trip with a stop in Peru for a summit of the Asia-Pacific Economic Cooperation forum (APEC). Among the leaders he is expected to meet there is President Xi Jinping of China.