By Brian Bariyo Tumuramye
Frenchman Koscielny, the 33, who had a medical on Tuesday, has returned to France after nine years with the Premier League club.
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 of the United States.
“We decided to sanction his move to Bordeaux once we agreed acceptable transfer terms,” Arsenal said in a statement.
“We thank Laurent for his contribution to the club and wish him all the best for the future.”
Former France international Koscielny rejected a new Gunners contract that he felt undervalued his contribution and continued ability.
Arsenal began disciplinary proceedings and instructed the captain to train with the under-23s, but offered him a way back into the first-team squad for the new season.
However, Koscielny had no intention of reversing his decision and was determined to secure a move to his home country, talks also took place with Bordeaux’s Ligue 1 rivals Rennes.
Arsenal hope to sign a new centre-back before the Premier League’s transfer deadline at 17:00 BST on Thursday, while they retain an interest in Celtic left-back Kieran Tierney despite the 22-year-old Scotland international being injured.
The Gunners are also in advanced talks over the sale of English right-back Carl Jenkinson, 27, to Championship side Nottingham Forest.
However, Arsenal have received no firm offers for defender Shkodran Mustafi and attempts to move the 27-year-old German on are further complicated by his desire to stay.
But foreign clubs have longer to negotiate a deal, depending on when their transfer windows close.
