The European leaders are expected to warn President-elect Mr Trump that now is not the moment to weaken sanctions against Russian Federation, which were imposed in 2014 over the conflict in Ukraine.
It opened with the burning question of the US-led coalition's battle against Islamic State jihadists in Syria and Iraq as well as Europe's migrant crisis, before delving into the Ukraine crisis, said an aide of Hollande about the talks that lasted just over two hours.
The leaders issued a joint statement pledging to work to secure "diplomatic resolutions to the conflicts in Syria and eastern Ukraine".
From Berlin, Mr. Obama will travel to Peru on Friday to attend a summit of Asian-Pacific leaders.
This week, Trump and Putin agreed in a telephone conversation that U.S.
But Obama urged the parties to focus on the whole, noting that "the achievements that we have seen on this continent in contrast to a divided Europe of the previous century, are ones that remind us of how important it is that we work together".
On Thursday, President Obama implied Mr. Trump would change his policy on some issues after a meeting with the German chancellor.
U.S. President Barack Obama called Merkel an "outstanding partner" during a joint news conference with the German leader on Thursday and said she might get his support if he were a German citizen and she chose to run again.
However observers said the recent seismic shifts in global politics could drive traditionally risk-adverse German voters back into her arms.
"I'm confident, although next year will be a hard one from an election point of view in Europe, that things will begin returning to normal", Rajoy said.
"We can't have a situation where people associated with certain groups now say, 'We are the people and everyone else isn't the people, ' " Merkel, who grew up in East Germany, told reporters.
The Prime Minister said the European leaders had also taken the "the opportunity to thank President Obama for the contribution he has made over the years and to wish him well for the future". Merkel cautioned on any bargains with Assad.
"He has brought untold suffering on his people, if you look at Aleppo and other places", she said.
Germany, which runs a budget surplus, has resisted pressure from European peers to spend more to help boost euro zone growth and last week the European Commission effectively urged Germany to loosen its purse strings next year.
"I think one will see that Mr. Trump will have his own political suggestions, but that the US won't be a different country tomorrow".
Angela Merkel made clear today that her top priority in Europe was the future of the European Union and not the details of Brexit as Theresa May flew in for talks in Berlin.
Also on Friday, NATO Secretary General Jens Stoltenberg gave a speech at an event hosted by the German Marshall Fund of the United States (GMF), where he said that Europe and the United States "are close economic and trade partners" and mentioned potential threats for the alliance.
"The biggest resistance is coming from the British", von der Leyen told the German weekly Die Zeit. "We remain committed to it and our partners remain committed to it as well".
Underscoring the heightened concern, European officials are speaking in increasingly stark terms. "And cyber-attacks", said Stoltenberg, listing the perceived dangers.