US President Barack Obama's Oval Office meeting on Thursday with his elected successor, Donald Trump, was an important first step for a smooth transition of power, the White House said.

“President Obama came away from the meeting with renewed confidence in the commitment of the president-elect to engage in an effective, smooth transition,” White House spokesman Josh Earnest told a news briefing.

Trump and Obama did not resolve their differences, Earnest said, but, based on their agreement on the need for an effective transition, “the meeting might have been at least a little less awkward than some might have expected.”

Earlier, Obama greeted Trump at the White House as the first public step in a transition of power after the Republican businessman's surprise victory in a bitterly fought election campaign.

There is no love lost between the two men, who have had almost no one-on-one contact previously. Trump led the “birther” movement that questioned Obama's US citizenship and has pledged to overturn the Democrat's signature policy achievements after he takes office on January 20.

Obama campaigned vigorously for Trump's Democratic rival, former Secretary of State Hillary Clinton, and called Trump both temperamentally unfit for the presidency and dangerously unprepared to have access to US nuclear codes.

Obama and Trump will seek to put their tensions behind them, at least for the cameras, during their Oval Office meeting.

Trump's motorcade took a South Lawn entrance into the White House, out of view of television cameras.

First lady Michelle Obama will also meet privately with Trump's wife, Melania, in the White House residence.

On Wednesday, Obama said that despite his major differences with Trump, he would follow the lead of former Republican President George W Bush in 2008 and ensure a smooth handover to Trump.

“Eight years ago, President Bush and I had some pretty significant differences, but President Bush's team could not have been more professional or more gracious in making sure we had a smooth transition,” said Obama.

“So I have instructed my team to follow the example that President Bush's team set,” Obama added.

After an unexpected election win on Tuesday that stunned the world, Trump spent Wednesday focusing on the transition during meetings with his staff at Trump Tower in New York.

While Democratic politicians in Washington were urging cooperation with the newly-elected president, anti-Trump demonstrations broke out in cities across the United States.

“Not my president,” shouted hundreds in New York. Demonstrators sat down on a highway interchange in Los Angeles blocking traffic and around 1,800 people in Chicago chanted “No Trump! No KKK! No racist USA” outside the Trump International Hotel and Tower.

Republican Chris Christie, who is leading Trump's transition team, said, “We heard a lot about the peaceful transfer of power during this election, and I think you'll see that symbolised today.”

During the campaign, Trump hinted that he might not accept the result if he lost to Clinton.

When asked whether Trump would apologise to the president for questioning his birthplace and legitimacy, the New Jersey governor, who could end up with a job in the Trump administration, said, "The controversy was just politics and they [Trump and Obama] have a lot more important things to talk about.”

Upon taking office, Trump will enjoy Republican majorities in both chambers of the US. Congress that could help him implement his legislative agenda and scrap or roll back Obama policies that he dislikes, such as the “Obamacare” healthcare law, the nuclear deal with Iran and US participation in the Paris agreement to fight global warming.

White House spokesman Josh Earnest said,"Obama would brief Trump about the benefits of those policies during their meeting."

Later on Thursday, Trump will hold separate meetings with the Republican leaders Paul Ryan and Mitch McConnell.

Ryan and Trump shared a strained relationship during the campaign, although they both ultimately said they supported each other. McConnell also kept a distance from Trump for most of the campaign.

Trump and Ryan will “discuss how they can hit the ground running in a Trump administration,”said a Republican source.

Trump's advisers are considering JP Morgan Chase & Co Chief Executive Jamie Dimon for the post of Treasury Secretary, CNBC reported, citing two people familiar with the matter.

The White House has laid out its plan to ensure a smooth transition, including giving representatives selected by Trump briefings on the work of US federal agencies.

"Trump and his senior aides will also start to receive daily briefings by US intelligence officials," said the White House .

The Obama administration also plans two “inter-agency exercises” for Trump's team aimed at handling and responding “to major domestic incidents.”

"The incoming US leader invited British prime minister Theresa May, the to visit as soon as possible," said British prime minister office.

"Trump aides were in touch with Russian government officials during the presidential campaign," said Russian Deputy Foreign Minister Sergei Ryabkov.