Klopp won the Bundesliga in 2011 and 2012 as Dortmund coach, with Grosskreutz in his squad, before moving to another traditional European club, Liverpool, where he lifted the Champions League and Premier League.
After leaving the Reds at the end of last season to recharge his batteries, it was announced this week that Klopp would become head of global soccer at Red Bull from January.
This would make him a top adviser to RB Leipzig, Salzburg, New York Red Bulls and the Austrian energy drinks firm's Brazilian and Japanese franchises.
Dortmund fans showed their disgust on social media, saying Klopp had abandoned his principles by joining a project which only began in 2005 - and Germany 2014 World Cup winner Grosskreutz agrees.
"Somehow it still hasn't really sunk in. I hoped it was fake news. It was a shock, sad in a way," Grosskreutz told the Viertelstunde Fussball podcast, an excerpt of which was put on Instagram on Friday.
"When we meet, I'll tell him personally that I think it's crap that he's doing something like this."
Grosskreutz, 36, was born in the city and is a lifelong Dortmund fan as well as an ex-player.
German media have speculated that Klopp is taking the reportedly lucrative Red Bull role as it does not involve intensive work on the pitch and leaves him available to one day succeed Julian Nagelsmann as Germany coach.
