Trump,70, flew from New York on his private jet and landed at Reagan National Airport, just outside the nation's capital. He broke from protocol and barred journalists from travelling with him to cover his first meeting with Obama.
The president-elect was accompanied by his wife, Melania, who will have a meeting with First Lady Michelle Obama. Vice President-elect Mike Pence also accompanied them.
The Republican president-elect has questioned Obama's US citizenship and vowed to dismantle his legacy. During the election campaign Obama called Trump "uniquely unqualified".
On Wednesday, Obama had urged all Americans to accept the result of the presidential election. "We are now all rooting for his success in uniting and leading the country," he said.
White House spokesman Josh Earnest has insisted that Obama is sincere about ensuring a smooth handover although he added: "I'm not saying it's going to be an easy meeting."