Senator Tom Cotton of Arkansas pointed to the dispute over Ms. Rice’s role in characterizing the 2012 attack on the diplomatic compound in Benghazi, which killed the American ambassador and three other American officials. “Susan Rice is the Typhoid Mary of the Obama administration foreign policy,” Mr. Cotton said on the Hugh Hewitt radio show. “Every time something went wrong, she seemed to turn up in the middle of it.”

Where Republicans detect scandal, Democrats see a smoke screen. Mr. Trump and his allies have been looking for indications of wrongdoing by Mr. Obama’s team for a month, since the president accused his predecessor of tapping telephones at Trump Tower during last year’s campaign. No evidence has publicly surfaced to substantiate that claim, and it has been widely dismissed. Having failed to prove the assertion, Mr. Trump and his allies have turned to Ms. Rice.

As the national security adviser, she received intelligence briefings six days a week. Last year, during the campaign, some of Mr. Trump’s associates were caught up in eavesdropping of foreign officials. When Americans who are not the target of a warrant are mentioned in reports about such surveillance, their identities are obscured, and they are typically referred to as U.S. Person One or U.S. Person Two.

But top officials, like the national security adviser, can ask the intelligence agencies to disclose the names, a process called unmasking. If the intelligence agencies agree, the briefer typically provides the identities orally.

Former national security officials have said that Ms. Rice was justified in asking for the names of Trump associates referred to in reports that intelligence agencies sent to her last year. The White House was concerned about attempts by the Russian government to interfere in the election, and she had an obvious need to be fully informed, they said.

Ms. Rice said Tuesday that the Russian meddling was a serious issue. “It was a grave concern to all of us in the national security team of the president and to the president himself,” she said. “We took this issue very seriously. We thought it was crucial to defend the integrity of our election process.”

She added, “For us not to try to understand it would be a dereliction of duty.”

Mr. Trump’s aides dismissed Ms. Rice’s comments. “Lyin’, leakin’ Susan Rice stammered through her soft ball interview with Dem PR person Andrea Mitchell,” Dan Scavino Jr., the White House social media director, wrote on Twitter.