The Gunners are due to ramp up preparations for the new campaign with a friendly against Colorado Rapids in Denver and three International Champions Cup encounters but will do so without their captain.
Koscielny's refusal to join his team-mates in making the trip comes amid intense speculation surrounding his future.
Ligue 1 sides Lyon and Bordeaux have been linked with the 33-year-old former France international, as have Bundesliga outfit Borussia Dortmund.
An Arsenal statement read: "Laurent Koscielny has refused to travel to the US for our pre-season tour.
"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 made 17 appearances in the Premier League last season after returning from a long-term Achilles injury in December.
His compatriot, teenage centre-back William Saliba, is on the cusp of completing a transfer to Emirates Stadium, but is expected to then return to Saint-Etienne on loan.