It is understood Arsenal blocked a possible move to Manchester United for Debuchy, who joined the Gunners from Newcastle for £12m in July 2014.
The France international, 30, has made just seven appearances for Arsenal this season and not figured since November.
The right-back has been allowed to return to France for the chance of first-team football ahead of Euro 2016.
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Arsenal defender Mathieu Debuchy has joined Ligue 1 side Bordeaux on loan for the rest of the season.