Griezmann, 35, has confirmed his departure to Major League Soccer side Orlando City when his contract expires in June, bringing the curtain down on a near-decade association with the club.
Sporting director Mateu Alemany is now working to identify his replacement, with Brandt emerging as one of the leading options.
The German international will leave Borussia Dortmund at the end of the season after Dortmund confirmed weeks ago that his contract, which expires in June, would not be extended.
He turns 30 in May and has been a regular starter this season, contributing 11 goals and three assists in 37 appearances. He has made over 300 appearances for Dortmund since joining in 2019.
Brandt's profile fits several of Atlético's requirements. He is comfortable operating as an attacking midfielder or on either flank, mirrors some of the positional flexibility Griezmann provided over the years, and arrives at no transfer cost.
Arsenal have also been linked with the player, who may also have interest from Major League Soccer. His preference is understood to be remaining in European football.