Donald Trump has spent most of the last decade seeking to undermine the legitimacy of President Barack Obama.

He was the force behind the so-called “birther movement” in which he tried to claim Mr Obama was born in Kenya, rather than the US, and was therefore not legally permitted to be its President. In turn, Mr Obama said Mr Trump was “uniquely unqualified” to become commander-in-chief.

So if the first meeting between the two men appeared a little cool when they spoke in the Oval Office on Thursday, it would have been no surprise. And when the President’s spokesman was asked whether they had discussed the issue of Obamacare - which Mr Trump has vowed to scrap on his first day - it was little surprise that he sought to dodge the question.

US President Barack Obama shakes hands with President-elect Donald Trump at the end of their meeting in the Oval Office of the White House in Washington, DC (EPA)

“I just had an opportunity to have an excellent and wide-ranging conversation,” Mr Obama said at the end of the meeting that lasted around 90 minutes, held without the presence of aides

“My number one priority in the coming two months is to try to facilitate a transition that ensures our President-elect is successful.”

He said that he was encouraged by Mr Trump’s desire to work with his team to ensure a smooth transition. He said it was important for the country “to now come together”.

U.S. President Barack Obama speaks while meeting with President-elect Donald Trump following a meeting in the Oval Office (Getty)

Mr Trump told reporters: “This was a meeting that was supposed to have last for 10 to 15 minutes. A chance to get to know each other, we have never met before. I have great respect. The meeting lasted for an hour-and-a-half.

“From my point of view it could have gone on even longer. We discussed a lot of different situations, some wonderful, some difficult.”

He said he looked forward to working again with Mr Obama, “including counsel”.

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Yet it remains unclear whether they discussed what must be among the many pressing concerns on Mr Trump’s mind - namely the future of one of Mr Obama's landmark pieces of legislation, the Affordable Care Act, better known as Obamacare.

Mr Obama’s spokesman, Josh Earnest, later declined to speculate as to whether the issue had been raised.

Trump calls President Obama "a very good man" as their first meeting in the Oval Office concludes

“I will let the President-elect read his record of the meeting,” he said.

Mr Earnest said the meeting had been called to deal with the issue of transition, rather than for the two men to narrow their ideological differences.

“When you consider the the political differences between the two men, I think the conversation was excellent,” he added.

Mr Trump travelled to Washington from New York for a first meeting with Mr Obama and senior officials to discuss the transition process that needs to be completed by January 20 2017. He also held a meeting with the Republican Party leadership. The relationship between Mr Trump and the party's elite has also been very strained.

This is a tradition for every new leader, and each department of the government has established so-called “landing teams”. These will work with members of Mr Trump’s team, headed by New Jersey Governor Chris Christie, as they prepare for the New Yorker to take office.