The 33-year-old centre-back is having a medical on Tuesday before completing a return to France after nine years with the Premier League club.
Frenchman Koscielny had a year left on his contract at Emirates Stadium and the Gunners wanted him to stay.
But he asked for a free transfer and refused to travel on a pre-season tour.
Former France international Koscielny rejected a new contract that he felt undervalued his contribution and continued ability.
As well as the medical, personal terms with Bordeaux still need to be finalised.