According to reports in Kicker, the 26-year-old's future at the Bay Arena is in doubt, as his excessive demands have halted discussions over a new deal.
Schalke and Hamburg are both keen on the Germany international, who is currently having treatment for a knee problem that has kept him off the pitch this season.
This uncertainty has led to speculation about current first-choice goalkeeper, Bernd Leno, who is on loan from Stuttgart.
The 19-year-old has affirmed that he wishes to return to his parent club, though Stuttgart's sporting director Fredi Bobic is not so certain.
"The contractual situation is clear, but we expect Bayer Leverkusen to talk with us in November or December," Bobic told Kicker. "We must always see what is the most sensible outcome for all concerned, for the clubs and the player. It is conceivable that he could be sold to Leverkusen, or a longer loan spell could be considered."
Adler was quick to dismiss rumors of a move, though.
"My only goal is a comeback," he said. "I want to be healthy. I have had no discussions with other clubs, not even with Schalke and HSV. The same goes for my agent."