A spokesperson for Rice later called the claim "ludicrous."
In the interview with The New York Times, Trump claimed that other Obama officials may have been involved in efforts to get the identity of, or "unmask," people with ties to Trump who were caught up in intercepted communications. The Times said Trump declined "repeated requests for evidence of his allegations or the names of other Obama administration officials."
The newspaper said that when Trump was asked if Rice had committed a crime, he replied: "Do I think? Yes, I think."
"I think it's going to be the biggest story," Trump told the newspaper, which he has repeatedly attacked for its coverage and called "failing."
In a statement to NBC News, a spokesperson for Rice said "I'm not going to dignify the president's ludicrous charge with a comment."
In an interview with MSNBC on Tuesday, Rice said allegations that the Obama administration used intelligence for "political purposes" were "absolutely false."