Jennings has spent the past couple of seasons with Bayern Munich after sealing a dream move to Germany from Tranmere Rovers in 2011.
He found it hard to settle, though, and put several English clubs on alert at the end of the season by revealing that he wanted to return to the UK.
Huddersfield Town are among the clubs interested in landing the 20-year-old but it is understood they are now closing in other targets and that Barnsley have now also entered the chase.
It is thought that the Tykes are optimistic of securing a deal for Jennings as they look to strengthen their squad ahead of the new season.
David Flitcroft's side narrowly avoided relegation from the Championship last term and will be looking to push on in 2013/14.