Langerak, who has made 27 appearances in all competitions in five seasons with the Bundesliga giant as perennial understudy to Ruben Weidenfeller, is said to have agreed terms with Stuttgart with only the formality of a medical standing in the way of an official announcement.
Reports in both Kicker magazine and Bild claim Langerak has decided to grab the chance of first team football elsewhere in response to the recent arrival of a third Dortmund keeper, Swiss internaional Roman Burki.
It’s understood that Burki, signed from relegated Freiberg, has been earmarked for the top spot at Dortmund by new head coach Thomas Tuchel ahead of Langerak, 26, and Weidenfeller, 34.
Stuttgart, which finished just one point off the relegation play-offs last season, is on the look out for a keeper after the departure this month of its first choice Sven Ulreich to Bayern Munich.
If the $4.3 million switch materialises, Langerak - who is currently the Socceroos back-up behind Mat Ryan - would compete with Polish interbational Przemysław Tyton for a starting role at Stuttgart.
Tyton spent last season on loan from PSV at La Liga side Elche.
Langerak would leave Dortmund with two Bundesliga titles, a German Cup and two German Supercups in his resume.