After nine years with the Gunners, Koscielny wants to return to his native France to see out his career.
But the defender’s unexpected decision to go on strike has caused outrage among Arsenal supporters.
Koscielny, who recovered from a ruptured achilles tendon last season, still has a year left on his contract at the Emirates.
And outspoken fan Morgan took to social media to lay into the former France international’s his lack of loyalty.
“WHAT?” the Good Morning Britain presenter tweeted in response to the news.
“Utterly disgraceful way for our Club Captain to behave.
“Get rid of him immediately. #afc”
Bordeaux have reportedly offered Koscielny a three-year deal.
Fellow Ligue 1 sides Lyon and Rennes are also interested in the 33-year-old, who has played 353 games for the Gunners.
Arsenal released a statement regarding the situation and are set to hit their captain with a fine.
“We are very disappointed by Laurent’s actions, which are against our clear instructions,” said Arsenal.
“We hope to resolve this matter and will not be providing any further comment at this time.”
Granit Xhaka would be the prime candidate to replace Koscielny as captain, should he be stripped of the armband.
Arsenal do not want to grant the centre-back a free transfer and will make him train with the under-23s while the club are in America.
The north Londoners are poised to sign William Saliba, 18, but they will loan the young defender back to Saint-Etienne next season.