WASHINGTON—A divide immediately arose Saturday between President Trump’s initial acceptance of Saudi Arabia’s response to the killing of journalist Jamal Khashoggi and senators in both parties who stepped up calls for sanctions and pressed the administration to do its own investigation.

“The story the Saudis have told about Jamal Khashoggi’s disappearance continues to change with each passing day, so we should not assume their latest story holds water,” said Sen. Bob Corker (R., Tenn.), who chairs the Senate Foreign Relations...