WASHINGTON—A Russian energy company partly owned by a friend of President Vladimir Putin has spent at least $280,000 in lobbying fees in the U.S. aimed in part at opposing a Senate bill that seeks to broaden U.S. economic sanctions against Russia for invading Ukraine, according to lobbying-disclosure records.

The payments, made by OAO Novatek to Washington lobbying and public-relations firm Qorvis MSLGroup over three months ended Sept. 30, provide a window into the behind-the-scenes struggle in Washington over Western economic...