LANGLEY, Va.—The President’s Daily Brief is among the U.S. government’s most closely held intelligence reports, its highly classified contents intended only for President Trump and a handful of his top national security aides.

Lately, an extra—and uninvited—pair of eyes has been looking in on the PDB, as it is known, even before Mr. Trump gets to peruse it. They belong to “Hudson,” a red-tailed hawk, buteo jamaicensis.

Many mornings at dawn—sometimes every morning over long stretches—the raptor has flown in and alighted on a railing just outside a seventh-floor corner office in the Central Intelligence Agency’s headquarters building here, a few corridors away from Director Gina Haspel’s executive suite.

Inside, a team of young intelligence analysts and their supervisors work through the night assembling the PDB and other classified reports from secrets supplied by CIA agents, the National Security Agency’s electronic intercepts, spy satellite images and other sources.

Hudson arrives just as the final touches are being put on the reports and as the spy agency’s briefers prepare to rush them to the White House and other locales around Washington for morning intelligence briefings.