WASHINGTON—Federal agents questioned then-National Security Adviser Mike Flynn in January, shortly after the White House denied he had talked about sanctions with a Russian official, according to people familiar with the matter.

The Federal Bureau of Investigation interview, which raises the legal stakes for Mr. Flynn and adds to the political pressure on the White House, came after the president’s spokesman Sean Spicer, at a Jan. 23 news briefing, said Mr. Flynn didn't discuss U.S. sanctions with Sergey Kislyak, Russia’s...