The 25-year-old Poland international looks likely to complete a free transfer to the Mercedes-Benz Arena after reportedly being offered a three-year contract
The Poland international made only four appearances for the club this season and is out of contract in the summer.
Stuttgart have reportedly offered a three-year deal to Boenisch, who rejected a proposal to extend his stay at the Weserstadion.
"It is true, I am in touch with this club," the 25-year-old admitted to Bild. "Stuttgart is a club that interests me, and the club will feature in European competition next season."
Stuttgart director Fredi Bobic confirmed: "Boenisch is one of the candidates we are taking a close look at. I can confirm this."
Boenisch is preparing for Euro 2012 and will be keen to wrap up a deal before the start of the tournament on Friday.