The Bosnia and Herzegovina international joins for a reported fee of 21 million euros (23 million dollars) plus up to 5 million euros in add-ons, which makes the deal both a record purchase for Stuttgart and a record sale for Augsburg.
Stuttgart, who needed to win a relegation playoff to stay in the Bundesliga in 2023, surprisingly finished second in the table last season, ahead of perennial champions Bayern Munich.
The second-place finish meant Stuttgart qualified for the Champions League for the first time since 2009-10.
Demirovic looks set to replace Serhou Guirassy, who scored 28 goals in 28 games for Stuttgart last season but has been linked with a move to Borussia Dortmund.
Guirassy had arrived in Dortmund but the transfer was postponed after a medical check. Dortmund announced on X, formerly Twitter, "an injury was discovered which requires further investigation."
Hamburg-born Demirovic scored 15 goals and laid on 10 assists in 33 games for Augsburg last season.
