WASHINGTON—Federal prosecutors are scrutinizing Turkey’s U.S. lobbying and the companies that backed a pro-Turkey initiative handled by President Donald Trump’s first national security adviser, according to people familiar with the matter.

In recent weeks, they say, a grand jury in Virginia has sought more information on efforts overseen by Michael Flynn’s private company to discredit a U.S.-based Turkish cleric. Turkey has accused the cleric, Fethullah Gulen, of orchestrating a failed 2016 coup attempt and has launched a...