Muto was born in Setagaya, Tokyo on July 15, 1992. When he played for Keio University, he joined J1 League club FC Tokyo in 2013. He played many matches from 2014. In summer 2015, he moved to German Bundesliga club Mainz. He played for the club in 3 seasons. In 2018, he moved to English Premier League club Newcastle United. In 2020, he moved to Spanish La Liga club Eibar.