There are still many issues to work out, especially because the Director of National Intelligence -- by statute -- has oversight over the agencies and is the President's prime contact on all intelligence issues, one of the sources said.
An administration official confirmed the move is under consideration.
Feinberg, a billionaire who leads the private equity firm Cerberus Capital Management, was a top economic adviser to Trump during the 2016 campaign and has been one of the President's friends for decades.
The New York Times first reported
the plans and said Feinberg has informed his company's shareholders he is in discussions to join the administration.
Trump has nominated former Indiana Sen. Dan Coats, a Republican, to be DNI but he has not been confirmed yet.
Before taking office, Trump aides denied reports that plans were in the works to revamp the Office of the Director of National Intelligence. But one of the officials who spoke to CNN on Thursday described the plan as an effort to consider possible reforms.
White House spokeswoman Sarah Huckabee Sanders told CNN that the White House doesn't "have any personnel announcements right now," but that the Trump administration is taking leaks "very seriously and we will be doing a thorough review."
Sanders would not say, however, that reporting on Feinberg's role is wrong.
"All I am saying is there is not an announcement at this time," she said.
Trump and his close aides have been bothered by the fact that a series of stories, including those illuminating the Trump's campaign's ties to Russia, have stemmed from the intelligence community.
"Leaking, and even illegal classified leaking, has been a big problem in Washington for years. Failing @nytimes (and others) must apologize," Trump tweeted on Thursday morning.
During a news conference with Israeli Prime Minister Benjamin Netanyahu Wednesday, Trump called leaks a "criminal action."
On Thursday, Trump told reporters: "We'll find the leakers. We're gonna find the leakers. They're gonna pay a big price for leaking."
Sen. James Risch, an Idaho Republican, said Thursday that he believes Trump is within his right to nominate someone like Feinberg to oversee this review.
"If the President feels that he needs to have a look at this and have someone advise him as to what the first branch of government is doing as far as the community operation is concerned that's certainly up to him," said Risch, who sits Senate Select Committee on Intelligence.