President Obama has described Donald Trump as ignorant, dangerous and unfit to lead the nation, but he’ll ask the Republican president-elect Thursday not to wipe out his cherished, legacy-making executive actions on everything from the Iranian nuclear deal to Obamacare.

In a meeting at the White House to prepare Mr. Trump for taking office, Mr. Obama will try to persuade the president-elect to honor what his aides call a “tradition” of preserving executive actions implemented by the departing chief executive.

“There is a tradition, particularly with regard to executive agreements of successive presidents preserving some element of continuity,” said White House press secretary Josh Earnest. “I don’t know whether or not that will apply in this case. But as a part of this effective, smooth transition, President Obama will have an opportunity to talk to President-elect Trump about some of the benefits of these policies.”

Mr. Trump campaigned on promises of repealing the Affordable Care Act, ripping up Mr. Obama’s international climate change agreement to curb carbon emissions and tossing out his free trade agreement with Pacific Rim nations and scaling back the executive order granting deportation amnesty to millions of illegal immigrants, among others. He has said his top priority is “to dismantle the disastrous deal with Iran” that lifted international economic sanctions in return for limits on Tehran’s nuclear programs.

The meeting will be Mr. Trump’s first visit to the White House since defeating Democrat Hillary Clinton early Wednesday in a bitterly fought campaign. After attacking Mr. Trump repeatedly as an intellectually lazy liar who wasn’t qualified to work at a 7-Eleven, Mr. Obama now finds himself in the position of seeking the Republican’s consent to protect many of the president’s achievements that were implemented unilaterally, without the approval of Congress.

Mr. Obama sought to extend an olive branch to Mr. Trump on Wednesday, first calling to congratulate the president-elect around 3:30 a.m. when the election results were finally clear, and later issuing a public call in the White House Rose Garden for national unity.

“Everybody is sad when their side loses,” Mr. Obama said with Vice President Joseph R. Biden at his side. “The day after, we have to remember we’re actually all on one team. We are now all rooting for his [Mr. Trump‘s] success in uniting and leading the country. We are Americans first.”

The president said he was encouraged by Mr. Trump’s remarks in his victory speech, and in their phone conversation, that the Republican will try to unite the country.

“We all want what’s best for this country,” Mr. Obama said. “That’s what I heard in Mr. Trump’s remarks … and I was heartened by that. That’s what the country needs — a sense of unity, a sense of inclusion, respect for our institutions, our way of life, the rule of law and respect for each other.”

White House officials also suggested that Mr. Trump, whose selling point with voters was his lack of government experience, will see the value of Mr. Obama’s policies after he gets some government experience. Mr. Earnest said White House officials believe the transition briefings will give Mr. Trump “a new appreciation for the benefits of those policies.”

“Am I suggesting that President Trump is going to reverse himself on a whole range of things that he’s been campaigning on for more than a year-and-a-half? No,” Mr. Earnest said. But he added that “there is a long tradition of presidents, even presidents in different parties, seeking to preserve some measure of continuity.”

The president’s spokesman also warned that Mr. Trump and his team will find out how difficult it is to govern if they try to unravel some of Mr. Obama’s signature achievements. He said repealing Obamacare would take away health insurance for about 22 million Americans, implying Mr. Trump would have 22 million angry voters on his hands.

While not outlining new executive actions and agreements the president might seek in the next two months, the White House also made clear that Mr. Obama will do everything in his power until he leaves office on Jan. 20 to keep intact the Iranian nuclear deal. Mr. Earnest said there would be international ramifications to the U.S. pulling out of the deal, such as the threat of Tehran restarting its nuclear weapons program.

“It’s not just as simple as some of the campaign rhetoric might make it seem,” he said. “The consequences, for example, with the Iran deal of pulling out, you do risk … the Iranians trying to break out. There’s also a U.N. Security Council Resolution that applies to this agreement. That means that this agreement is something that’s supported by our allies, but also by Russia. That I think could be a pretty good indication of how united the international community is behind this agreement.”

Mr. Obama and Mrs. Clinton accused Mr. Trump during the campaign of cozying up to Russian President Vladimir Putin and essentially ignoring evidence that Moscow was supporting cyberattacks against the Democratic Party and U.S. elections systems.

White House aides were clearly still in shock Wednesday over Mr. Trump’s victory and its implications for wiping out Mr. Obama’s priorities of the past eight years. After Mr. Obama spoke in the Rose Garden to hundreds of staffers, many left in tears.

After portraying Mr. Trump during the campaign as a dishonorable whiner whose election would cause America to “spin out of control,” Mr. Obama was unapologetic about his harsh rhetoric.

“It is no secret that the president-elect and I have some pretty significant differences,” Mr. Obama said. But he added that he also had sharp differences with his predecessor, Republican George W. Bush.

“President Bush’s team could not have been more professional, or more gracious, in making sure we had a smooth transition,” Mr. Obama said. He said he’s instructed his team to follow Mr. Bush’s example, “to make sure that this is a successful transition for the president-elect.”

Mr. Earnest said Mr. Obama hasn’t changed his negative views about Mr. Trump, but feels a duty to make the transition as smooth as possible.

“The president has no desire or inclination to paper over the deep differences that the two of them have,” he said. “I’m not saying it’s going to be an easy meeting, but the president is deeply sincere about fulfilling this responsibility.”

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