Republican Trump had vowed to jail Clinton and crowds at his rallies chanted "Lock her up." But he now believes she "has been through enough," MSNBC reported on Tuesday, citing an unidentified source.
Addressing the report in an interview with MSNBC, senior Trump advisor Kellyanne Conway indicated it was correct.
"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," she said on MSNBC's "Morning Joe" program.
During the campaign, opinion polls found that many Americans did not find Clinton trustworthy.
The FBI investigated Clinton's use of a private email server while she was secretary of state during President Barack Obama's first term, concluding in July that her actions were careless but that there were no grounds for bringing charges.
The Clinton Foundation charity has also been scrutinized for donations it received, but there has been no evidence that foreign donors obtained favors from the State Department while Clinton headed it.
If Trump does abandon his vow to appoint a special prosecutor for Clinton it will be a reversal of a position he mentioned almost daily on the campaign trail, when he dubbed her "Crooked Hillary."
The right-wing news website Breitbart accused the New York businessman of breaking his word. "Broken promise: Trump 'doesn't wish to pursue' Clinton email charges," the site said on its front page on Tuesday. Breitbart was run until earlier this year by Steve Bannon, who became a leader of Trump's election campaign and has now been named to be his chief strategist when he takes over the White House on Jan. 20.
Judicial Watch, a conservative group that has vigorously pursued probes into Clinton, also urged Trump not to drop his promise to investigate her.
Trump has been holding meetings since his Nov. 8 election victory to fill senior posts for his administration before he takes office. Conway said he had moved on from the campaign.
"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," Conway said.
Trump issued a video message on Monday about the transition and his plans for the start of his term, but he did not mention important campaign positions such as his promises to scrap the Iran nuclear deal and to repeal Obama's healthcare reform.
Instead, Trump vowed to pull out of the Trans-Pacific Partnership, or TPP, trade deal, and to do away with energy regulations.
Congress can pursue its own investigations of Clinton, regardless of whether Trump opts to appoint a special prosecutor.
U.S. Representative Jason Chaffetz, Republican chairman of the House Oversight and Government Reform Committee, has said he will continue investigating Clinton's use of a private server.
But Conway indicated Trump would frown on that.
"When the president-elect, who's also the head of your party now, tells you before he's even inaugurated he doesn't wish to pursue these charges, it sends a very strong message - tone and content - to the members," Conway said on MSNBC.
Clinton has said her use of a private email server was a mistake and has denied links between donors to her family's foundation and her work as secretary of state.
Trump was to meet on Tuesday with journalists from The New York Times, a spokeswoman said, hours after he posted on Twitter that he was canceling the meeting.
Trump, who has an adversarial relationship with the newspaper, has not held a news conference to talk about his priorities since his election.
On Monday, he met with television anchors and news industry executives and reporters in a session The Washington Post described as a contentious but generally respectful.
Trump singled out reporting of his campaign by CNN and NBC that he considered to be unfair, the Post said, citing four participants at the meeting in New York.
Trump, who has never previously held public office, was quick to bristle at unflattering news coverage during the campaign, even as he remained accessible to certain reporters, including several from the Times.