And it appears he is softening his stances on some more campaign pledges, too.
It comes as Mr Trump has condemned a white nationalist group that celebrated his election win with Nazi salutes.
United States president-elect Donald Trump has made repeated promises to prosecute rival Hillary Clinton over her use of a private email server while she was secretary of state.
It was a line he relied on often throughout the bitter and divisive election campaign.
"I am going to instruct my attorney-general to get a special prosecutor to look into your situation." ... "She's as crooked as they come. She had a little bad news today." ... "They are reopening the case into her criminal and illegal conduct." ... "Let's knock out Crooked Hillary Clinton, crooked as a $3 bill." ... "The most corrupt person ever to seek the office of the presidency of the United States." ... "And we can all go against Crooked Hillary Clinton and beat her, really, really beat her big-league."
That core theme on the Trump campaign trail often led supporters at rallies to chant, "Lock her up."
But now, Mr Trump is apparently backing off on the issue.
His senior adviser, KellyAnne Conway, has indicated he will not push for further investigations into Ms Clinton's email practices nor the often-targeted Clinton charity foundation.
"I think Hillary Clinton still has to face the fact that a majority of Americans don't find her to be honest or trustworthy, but, if Donald Trump can help her heal, then perhaps that's a good thing. I do ... look, I think he's thinking of many different things as he prepares to become the president of the United States, and things that sound like the campaign aren't among them."
Former New York mayor Rudy Giuliani, a vice chairman of Mr Trump's transition team, says he will support Mr Trump's decision either way.
"If he made that decision, I would be supportive of it. I'd also would be supportive of continuing the investigation. I think the President-Elect had a tough choice there. You could go either way. If he made the choice to unite the nation, I think all those people who didn't (did) vote against him maybe can take another look at him."
And it may not be the only position Mr Trump is backing down on.
The President-Elect appears to have shifted his stance on climate change, which he called a hoax in the lead-up to the election.
After meeting reporters in New York, Mr Trump reportedly told a journalist with The New York Times he thinks there is "some connection between climate change and human activity."
It comes after a brief Youtube video address in which Mr Trump outlined his priorities for his first hundred days in office.
"My agenda will be based on a simple core principle: putting America first. Whether it's producing steel, building cars or curing disease, I want the next generation of production and innovation to happen right here on our great homeland, America, creating wealth and jobs for American workers."
His tactic to deliver his agenda via social media goes against the traditional practice of holding a press conference in the wake of an election.
Mr Trump has not held a formal press conference since the election.
And his senior adviser, Ms Conway, has indicated no plans for one in the near future.
"The man works 18 hours a day, interviewing people, taking calls from across the world, and he will have a press conference in due course."
Meanwhile, the President-Elect has condemned a white nationalist group that cheered his election win.
A video released by Atlantic Monthly magazine shows members of the National Policy Institute giving Nazi salutes in celebration of Mr Trump's election victory.
"Hail Trump! Hail our people! Hail Victory!"
That is the group's president, Richard Spencer, who, in the video, delivers remarks that include the term 'lugenpresse' which, in German, means "lying press."
"The mainstream media, or perhaps we should refer to them in the original German, 'lugenpresse' ..."
The phrase rose to infamy during the Nazi era.
The deputy national director of the Anti-Defamation League in the United States, Ken Jacobson, says he is both "shocked and not shocked" by the video.
"I say shocked, because, so close to the halls of power, to see that kind of expression, open support for racism, anti-Semitism, Nazis, it's still an amazing thing. I say not shocked only because, working for the Anti-Defamation League, we monitor such groups, and we know these things exist. What's changed is a sense of legitimacy that they feel they have gotten from all that's taken place during this campaign."
