Jennings, who joined Bayern two years ago from Tranmere Rovers, left the European champions earlier this summer after failing to settle in Germany.
Bayern granted Jennings a free transfer despite the player impressing during his time at the Allianz Arena.
A number of English clubs expressed an interest in signing the skilful winger, but it appears Barnsley have won the race for his signature.
The 20-year-old is now set for talks and a medical at Barnsley as Tykes boss David Flitcroft looks to add the youngster to his squad.
Jennings' signing would be a major coup for Barnsley as they look to push on next season after avoiding relegation last term.