President Trump egged on scrutiny of President Obama’s national security adviser Tuesday by re-posting a story that calls into question Susan E. Rice’s alleged role in “unmasking” campaign aides who were swept up in intelligence monitoring.

Mr. Trump retweeted a story by The Daily Caller that had been posted to the Drudge Report website under the headline: “RICE ORDERED SPY DOCS ON TRUMP?”

The story claims that Ms. Rice asked U.S. spy agencies to produce “detailed spreadsheets” of phone calls involving Mr. Trump and his aides during the campaign, citing former U.S. Attorney Joseph diGenova as its source.

Ms. Rice played a central role in “unmasking” several Trump campaign officials who had been swept up in U.S. surveillance operations against foreign targets during last year’s presidential election campaign, according to current White House officials and sources on Capitol Hill.

They said Tuesday that Ms. Rice requested that names be provided for otherwise unidentified U.S. people in dozens of raw intelligence reports relating to the Trump campaign.

While Ms. Rice’s actions and alleged interest in the Trump campaign appear to have been within her legal authority as national security adviser, the potentially explosive revelation has touched a nerve in Washington and stirred speculation that she could be called to testify on Capitol Hill about Russian election meddling.

The president’s son — Donald Trump Jr. — fanned the flames on Twitter by hailing Mike Cernovich, a documentary filmmaker, for sounding the alarm over Ms. Rice’s alleged role.

“Congrats to @Cernovich for breaking the #SusanRice story. In a long gone time of unbiased journalism he’d win the Pulitzer, but not today!” he tweeted.

Sen. John McCain, Arizona Republican, told CNN the calculus should be whether Ms. Rice acted within her security role or flouted her duty for political purposes, if the reports are confirmed by investigators.

But Sen. Rand Paul, Kentucky Republican, has seized on the reports as a big deal, saying it confirms his fear that too many Americans are being swept up on surveillance of foreign actors without a traditional warrant.

“I think it’s am mistake to downgrade this and say, ‘Oh is was just incidental and it’s no big deal,’” Mr. Paul told MSNBC.

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