The Austrian has been impressing while on a season-long loan at the 2. Bundesliga club, who are in fifth place and challenging for promotion to the top flight.
Kershbaumer, 25, has scored four goals in 14 league appearances for Bielefeld.
But Brentford have told the German club that they will have to pay half a million euros for the player, who is under contract at Griffin Park until 2019, if they want to turn the loan into a permanent deal.
That could be a problem unless they can achieve promotion, but sporting director Samir Arabi intends to push the boat out if they can.
“I do not talk about speculation,” he told Bild. “But we will fight and pull all the stops to keep Kerschbaumer.”