Sky Sport Germany confirmed that Galatasaray have already made contact with Brandt's representatives and are described as very interested in bringing him to the Süper Lig champions.
Enquiries have also arrived from MLS clubs and a Japanese outfit, but Brandt has no interest in either destination. He is adamant about continuing at Champions League level.
Spain remains his clear preference. Atletico Madrid, who must rebuild their attacking line following the departure of Antoine Griezmann to Orlando City, have made direct contact with Brandt's family and a meeting is expected to follow the conclusion of the Bundesliga season. Discussions with RCD Mallorca earlier in the year came to nothing and will not be revived.
Brandt told Sky Sport Germany at his farewell home game: "I'm taking it one step at a time. It won't drag on until August."
He leaves Borussia Dortmund having scored 57 goals and provided 70 assists in 307 competitive appearances across seven seasons at the club. He won 48 caps for Germany before falling out of Julian Nagelsmann's plans and will not feature at the World Cup.
Brandt's profile as a free agent with extensive Champions League experience, technical quality and fluency in Spanish makes him an attractive proposition. Barcelona have been loosely linked, while Arsenal are also said to be monitoring the situation. Atletico's sporting director Mateu Alemany is understood to be leading the approach.
A decision is expected within weeks.