BILD says Gladbach have searched for replacements for Thuram, Italian broadcaster Sport Mediaset have claimed Bayern Munich have offered a 'lucrative' five-year deal for the forward.
Unlike Arsenal, Bayern are unable to seal a deal next month due to transfer rules. The Gunners, from January, are open to begin talks with Thuram once the January window opens over a free transfer in January while any German sides have to wait until the summer.
Monchengladbach coach Oliver Neuville has already admitted that the difficulty of keeping Thuram, calling the chances of the club retaining his services as 'slim'.
He said: "He's having an amazing season. He's scored a lot of goals. I think it will be difficult for him to stay at Gladbach," he told Radio Sportiva.
"If he continues to score so many goals, it's certain he'll have many offers from Italy, Spain and England. We must also be a little realistic. I hope he will still be here for a long time in Monchengladbach. But now he's been called up by French for the World Cup, it will be difficult for him to extend. I hope (he stays). But the chances are slim."