The 29-year-old is currently a free agent and Stuttgart's director of sport Fredi Bobic is pondering offering him a trial.
"I am in contact with Andi and there is a good chance that he will keep himself fit with us and train with the first team," Bobic told the Bild newspaper.
"I will discuss it with (coach) Bruno Labbadia and then we will see if it makes sense."
Stuttgart was Hinkel's first club as a professional and he made his debut in the UEFA Cup tie with Celta Vigo in 2001, spending six seasons with the club.
He then moved on to Sevilla in 2006 before joining Celtic in 2008.
He won the last of his 21 Germany caps in a friendly against the United Arab Emirates in June 2009.
"You can never rule anything out in football," said Babic. "But we want to be fair with Andi and will therefore not be making him any promises."