Activision Blizzard Inc. Chief Executive Bobby Kotick has told senior managers he would consider leaving the company if he can’t quickly fix the culture problems at the videogame giant, according to people familiar with his comments.

Mr. Kotick, who has led Activision for three decades, stopped short of saying he would step down in a Friday meeting with executives of the company’s Blizzard Entertainment unit, but left the possibility open if misconduct issues across the company weren’t fixed “with speed,” these people said.