Koscielny has been desperate to return to France this summer and refused to go on Arsenal’s pre-season tour of the United States in a bid to force a move.
The club captain had hoped that Arsenal would allow him to leave for free, despite having one year left on his current contract, but the club were adamant that they would only countenance his departure if they received a fee.
Koscielny is due to have a medical with Bordeaux on Tuesday and is believed to have already agreed personal terms with the French side.
Unai Emery, the Arsenal head coach, had tried on multiple occasions to convince Koscielny to stay in north London.
The defender turns 34 in September, but the coaching staff believed he could still be a crucial player this season before the arrival of teenage centre-back William Saliba, who signed this summer but has been loaned back to France, at the end of the campaign.
Arsenal are continuing to pursue another centre-back before the transfer deadline day on Thursday. There is interest in Juventus centre-back Daniele Rugani, but it is understood that the financial figures being quoted by the Italian club are out of Arsenal’s reach.
Reports in Italy have said that Juventus want around £50m for the 25-year-old, who has made seven appearances for the Italian national side.
Meanwhile, Arsenal are in talks with Nottingham Forest over the departure of Carl Jenkinson. The right-back made just three appearances in the Premier League last season and is not part of Unai Emery’s first-team plans going forward.