The French central defender failed to board the flight which left on Thursday, after reportedly informing manager Unai Emery the previous day of his feelings.
The club responded by issuing an angry statement on their official website, citing their disapproval at Koscielnys behaviour.
"We are very disappointed by Laurent’s actions, which are against our clear instructions," the statement began. "We hope to resolve this matter and will not be providing any further comment at this time."
It had been thought that Koscielnys decision was based around his desire to move back to France, with Arsenals captain said to have believed that an agreement was in place for him to be released from his contract this summer when the club failed to qualify for the Champions League back in May.
Club statement on Laurent Koscielny
— Arsenal FC (@Arsenal) July 11, 2019
However, BBC Sport journalist David Ornstein claims that it is not just a desire to return to his homeland that caused Koscielnys no-show.
Arsenal have signed just one player so far during the transfer window, the 18-year-old Brazilian Gabriel Martinelli. This has left Koscielny questioning the clubs ambition, as they prepare for their third straight year in the Europa League next season.
Moves for Wilfried Zaha and Kieran Tierney have also been rejected, meaning that Arsenal are set to enter the new season with a similar squad to that which fell short domestically and in Europe last year.
The 33-year-old is also said to be frustrated by the way he was rushed back in to action last season following a long-term achilles injury. Koscielny was forced to miss Frances triumphant World Cup campaign in 2018 with the issue, but was brought back in to the Arsenal side in December 2018 after a swift recovery.
Yet he played twice in three days upon his return, a feat that he was asked to do a further five times in the remainder of the season due to Arsenals lack of defensive options. The experienced defender is believed to feel that this was an unnecessary risk, leading to further friction between himself and the manager.