The full-back has lost his place in both sides and may seek a move away in January with Euro 2016 looming.
The defender has found himself down the pecking order at the Emirates in recent times, with youngster Hector Bellerin taking his place while he was out injured last season.
The young Spaniard has continued to impress in the position this term as well, with Debuchy having spent the majority of the first seven games of the campaign on the bench.
That has led to his omission from Didier Deschamps' squad for France's forthcoming fixtures, leaving the full-back fearful he will miss out on the European Championships next summer if he remains out of the picture in north London, reports the Daily Mail.
With Bellerin looking solid despite his young years, and Wenger clearly sticking with him so far, a move in January may be the only choice for Debuchy.
There is sure to be interest is his services as well, with a number of clubs likely to see him as a sensible purchase should the Gunners be willing to sell in the next transfer window.
Until then, the Frenchman has a fight on his hands to try and win back his position in the Arsenal team, as well as the France fold.