After injury cut his campaign short last season, the 29-year-old has made just two appearances in the Bundesliga this term, seeing Marco Reus take his place in attacking midfield.
"I want to solve the current situation as soon as possible," he told Nikkan Sports. "One way is to change the team. I want to change the environment and the challenge.
"I have played in Dortmund and Manchester United, no matter what country I go to, I will not be afraid.
"I have always been thinking about how I can go to Spain in the last couple of years. I can't retire without going to Spain.
"This is the most important challenge in my career."
Kagawa has made 216 appearances for Dortmund over his two spells at the club and is contracted to the Bundesliga side until 2020.
He is not the only experienced player at Dortmund to find playing time difficult to come by, with Mario Gotze featuring just five times in the league.
Dortmund are enjoying a fine season and sit comfortably at the top of the Bundesliga with 30 points from 12 matches and are yet to lose in the top flight.
Coach Lucien Favre has received significant praise for putting trust in young players, with the likes of Jadon Sancho, Achraf Hakimi and Christian Pulisic flourishing.