The 18-year-old, who moved to North London two years ago, has caught the eye for the Gunners this season and got his first goal in their recent 2-1 win over Swansea City.
"It's true, he had the chance to return in the summer but he opted to fight for his chance at Arsenal and deserves every respect for this decision," Schneider told Bild. "After his goal in Swansea I sent his father a text to congratulate him.
"We will continue to monitor his progress, but if he continues to go down the path he is currently going and becomes a starter for Arsenal, it will get difficult."
Gnabry made his debut for Arsenal in the League Cup against Coventry City last year.