Grosskreutz, a World Cup winner with Germany, signed for the Turkish champions in the summer but has yet to play for the club as his move from Borussia Dortmund was completed after the transfer window deadline in Turkey.
Fifa subsequently denied Gala permission to play Grosskreutz until January, but rumours soon circulated that he was unsettled in Istanbul.
Speculation gathered pace that he could reunite with former Dortmund coach Jurgen Klopp at Liverpool, but the 27-year-old now looks set to return to Germany's top flight with Stuttgart.
"Kevin brings excellent footballing qualities and an absolute winning mentality," sport executive Robin Dutt told the club's official website.
"He can be used in various positions and would be an enormous gain for our squad. After the unfortunate time in Istanbul, Kevin is eager to finally play football."
Nevertheless, the official also stressed that a deal is not yet finalised, stating: "The agreement with Galatasaray on a transfer is pending.
"We hope that the agreement is carried out promptly."
Stuttgart also confirmed that a loan deal has been agreed for Dynamo Kiev striker Artem Kravets, although there are still formalities to finalise on that transfer as well.