Stuttgart claim they had been looking to take Gnabry back to his homeland prior to his breakthrough in north London, but accept that the 18-year-old may now be out of their reach.
Schneider told BILD: "It is a fact that he had the chance to return to our club - again over the summer.
"He decided to stay on at Arsenal and try to force his way into their first team. Everyone should respect the decision Serge Gnabry has taken.
"We are still monitoring him and his progress. But there is little doubt that it would be very difficult to re-sign him in the future if he keeps up the good work and turns into an Arsenal regular."
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