Seven years ago, a political appointee in the first Trump administration approached U.S. immigration authorities with a request that alarmed career staffers: Scrub the immigration records of Rep. Ilhan Omar, then a newly elected Democratic lawmaker from Minnesota, for evidence of fraud.

That 2019 demand, detailed by three former immigration officials, struck staffers at the time as a highly unusual effort to wield federal power to dig up compromising material on a sitting member of Congress who was a prominent critic of President Donald Trump.

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The request sparked a swift review examining Omar’s history, her family’s past as Somali refugees resettled in the United States and rumors that had circulated for ...