US President-elect Donald Trump thanked President Joe Biden for pledging a smooth transfer of power as the victorious Republican made a historic return visit to the White House on Wednesday.

"Politics is tough, and in many cases it's not a very nice world. It is a nice world today and I appreciate it very much," Trump said after the two men shook hands in the Oval Office.

Trump, 78, added that the transition would be "smooth as you can get."

Biden greeted Trump in front of a roaring fire, offering him congratulations and saying: "Welcome back."

The 81-year-old Biden invited his sworn rival to the White House –despite the fact that Trump, who refused to admit his 2020 election loss, never afforded Biden the same courtesy.

Biden, who dropped out of the election in July but saw his successor Kamala Harris lose to Trump last week, said he was "looking forward to having a smooth transition."

"We'll do everything we can to make sure you're accommodated, (have) what you need," he told Trump.

Ahead of the White House visit, Trump addressed Republicans from the House of Representatives at a Washington hotel near the Capitol, which a mob of his supporters stormed in 2021 to try to reverse his election loss.

An ebullient Trump suggested that he could even be open to a third term in office – which would violate the US constitution.

"I suspect I won't be running again unless you say, 'He's good, we got to figure something else,'" he said, drawing some laughter.

Trump's party looks set to take both chambers of Congress and consolidate his extraordinary comeback.

He was accompanied at the meeting with Republicans by the world's richest man Elon Musk, whom he named on Tuesday as head of a new group aimed at slashing government spending.

Trump has launched a flurry of nominations as he moves swiftly to name his administration.

In the latest announcement, Trump said he would be bringing four advisers from his presidential campaign into his new White House as deputy chiefs of staff, including two longtime aides and the political director who played a key role in his victory.

The incoming president said longtime aide Dan Scavino will serve as a deputy without giving a specific portfolio, campaign political director James Blair as deputy for legislative, political and public affairs, and Taylor Budowich as deputy chief of staff for communications and personnel.

All will have the rank of assistant to the president.

Trump also formally announced Stephen Miller, an immigration hard-liner, will be deputy chief of staff for policy and homeland security adviser. That had previously been confirmed by Vice President-elect JD Vance on Monday.

Blair was the political director for Trump’s campaign and, once Trump became the presumptive GOP nominee, the political director for the Republican National Committee.

Scavino was a senior adviser on Trump’s campaign and, in his first term in the White House, he worked as a social media director.

He began working for Trump as a caddy at one of Trump’s golf courses, and was part of the small group of staffers who traveled with the president across the country for the entirety of the campaign.

Before joining the campaign, Budowich worked for the pro-Trump Super PAC, Maga Inc., and after Trump left office, Budowich served as his spokesman while working for Trump's political action committee, Save America.

“Dan, Stephen, James, and Taylor were ‘best in class’ advisors on my winning campaign, and I know they will honorably serve the American people in the White House,” Trump said in a statement.

“They will continue to work hard to Make America Great Again in their respective new roles.”

Miller is one of Trump’s longest-serving aides, dating back to his first campaign for the White House. He was a senior adviser in Trump’s first term and has been a central figure in many of his policy decisions, particularly on immigration, including Trump’s move to separate thousands of immigrant families as a deterrence program in 2018. (Agencies)