The Gunners released a statement on Thursday lamenting the action and highlighted their disappointment at the 33-year-old Frenchman, who has been at the club since 2010.
“Laurent Koscielny has refused to travel to the US for our pre-season tour,” said Arsenal.
“We are very disappointed by Laurent’s actions, which are against our clear instructions.
“We hope to resolve this matter and will not be providing any further comment at this time.”
Koscielny, who is reportedly keen on returning to France, was said to have asked to be released from the final year of his contract.
He has played 353 times for the North London side, winning the FA Cup twice, and was named club captain at the start of last season following the retirement of Per Mertesacker.