WASHINGTON—FBI Director James Comey is back in the crosshairs of Democrats, who say he hasn’t been forthcoming with congressional investigators probing alleged Russian interference in the U.S. presidential election, a situation that could jeopardize their ability to study whether the Kremlin colluded with anyone connected to President Donald Trump.

After a three-hour, closed-door briefing with Mr. Comey on Thursday, Democrats on the House Intelligence Committee emerged to say that the Federal Bureau of Investigation director...