The Ligue 1 side bought Laurent Koscielny earlier this summer after the centre-back refused to travel to the United States for the Gunners' pre-season tour.
And they are now plotting a return to the Emirates Stadium as they seek to bolster their midfield options.
Bordeaux have been attempting to prise Youssef Ait Bennasser away from domestic rivals Monaco in recent weeks.
However, negotiations have been difficult and les Girondins are poised to turn their attention to Elneny.
Unai Emery has reportedly told the Egypt international that he is free to find another club before the European transfer window closes next Monday.
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