Trump says in the interview he thinks the Susan Rice allegations will be "one of the big stories of our time". "It's such an important story for our country and the world".
Trump told the New York Times Wednesday that media outlets, including the Times, have not adequately covered the story.
The president earlier told The New York Times he thinks Rice committed a crime.
Trump was unclear what crime specifically he is accusing Rice of committing. While Rice would not deny that she asked that names of Trump officials be demasked, she insisted the Obama administration did not spy on Mr. Trump or his staff for political purposes.
The U.S. official said Trump's National Security Council began an intelligence policy review after the inauguration that included looking into how widely raw intelligence and the names of Americans were disseminated.
"The charges [against Nunes] are entirely false and politically motivated, and are being leveled just as the American people are beginning to learn the truth about the improper unmasking of the identities of US citizens and other abuses of power", he said.
Americans' names also can be included if doing so "is necessary to understand foreign intelligence information or assess its importance", according to guidelines made public by the Office of the Director of National Intelligence, which oversees the 16 USA intelligence agencies.
"The allegation is that somehow the Obama administration officials utilized intelligence for political purposes", she said Tuesday.
Majority members on the House and Senate committees could call Rice, or other people from her era, to testify as Democrats seek Trump campaign aides.
In mid-March, House intelligence committee chairman Devin Nunes abruptly announced he had seen "troubling" information about spy agencies widely spreading the identities of Trump associates.
The president provided no evidence to back his claim.
Rice has not confirmed or denied making the requests, but denies that any requests were politically motivated.
Intelligence officials said any requests that Rice may have made would have been handled by the intelligence agency responsible for the report, which in most cases would have been the National Security Agency.
White House spokesman Sean Spicer would not comment on the Rice development but did criticize the press corp for their coverage.
There is no evidence that Rice improperly requested that names be unmasked.
Nunes also said he confirmed that "additional names of Trump transition team members were unmasked".
The U.S. routinely monitors the communications of foreigners.
An April 3 Daily Caller report that Rice ordered US spy agencies to produce "detailed spreadsheets" of legal phone calls involving Donald Trump and his aides when he was running for president makes this story more interesting. "I never have and never will". However, senior national security officials can request the identities of the Americans if that helps understand the intelligence better. In an exclusive interview with MSNBC's Andrea Mitchell Monday, April 4, Rice flatly denied allegations she tried to "unmask" Trump campaign officials caught on surveillance by USA intelligence services with ill intent.