Rice declined to comment through a spokeswoman: "I'm not going to dignify the President's ludicrous charge with a comment."
Trump, his aides and other Republicans have suggested that Rice requested the information for political purposes and leaked them in news media reports. This week on Twitter, Trump called attention to a link on the Drudge Report, a conservative website, that described Rice as having ordered "spy docs" on him.
In an interview yesterday, Rice denied the allegations.
"I leaked nothing, to nobody, and never have and never would," Rice said on MSNBC. "The allegation is that somehow Obama Administration officials utilised intelligence for political purposes."
"Absolutely false," she added.
Rice may now face an inquiry by the Senate Intelligence Committee, which is looking into Russian interference in the 2016 election. The committee's chairman, Senator Richard Burr, said that Rice may be interviewed as part of the committee's work.
Rice has said that the pace of intelligence reports on Russian meddling increased beginning in the northern summer of 2016.
Trump's comments also pointed to a larger conspiracy, that would include other Obama Administration officials.
There is no evidence that Rice improperly requested that names be unmasked.
House Intelligence Committee Chairman Devin Nunes first suggested that he had seen intelligence reports that contained the identities of Trump associates, including some where the names were masked but were easily identifiable.
Those reports, which Nunes revealed in a news conference and were the foundation for a briefing he provided to the president, were uncovered by National Security Council officials working in the White House who, the Washington Post reported, secretly passed them on to Nunes.
Trump has also claimed, without evidence, that Obama ordered a wiretap of Trump Tower during the election. Democrat and Republican lawmakers continue to say no evidence supports this claim.