Former national security adviser Mike Flynn is cooperating with the probe of Russian meddling in the 2016 election, pleading guilty Friday to lying about calls with Moscow’s ambassador a month before President Trump’s inauguration, contacts that prosecutors said were coordinated with senior members of the president’s transition team.

Jared Kushner, the president’s son-in-law, was one of those advisers who spoke with Mr. Flynn about at least one of the calls, said people familiar with the matter, bringing the long-simmering...