WASHINGTON—Sen. Mark Warner, the top Democrat on the Senate committee investigating Russia’s interference in the U.S. election, said he plans to introduce legislation that would make political advertising on social media more transparent.

The proposal, which he is writing with Sen. Amy Klobuchar (D., Minn.), comes in response to recent revelations that Russian-tied entities bought divisive political ads on social-media platforms during the campaign season. Mr. Warner said the ads were intended to “sow chaos and drive division”...