Despite suffering for form at times, Xhaka is one of few standout options given the lack of leaders at Arsenal following the departures of veteran goalkeeper Petr Cech and long-term servant Aaron Ramsey.
Koscielny is their longest-serving current player but his exit now appears to be a formality.
It is claimed they were willing to offer him an extra year on his current £120,000-a-week deal.
However, the report from the Mirror adds they wanted to reduce his basic pay and make it up with performance-related bonuses while retaining an option to release him next summer.
He is said to be unhappy with terms and now desperate to force his way out.
Koscielny reportedly returned to the club on Monday but refused to train with the rest of his teammates.
Having failed to travel to America, Arsenal are now forcing him to report to the club every day to train with the under-23s.
The club issued a club statement on Thursday morning after the reports of the rift emerged.
“We are very disappointed by Laurent’s actions, which are against our clear instructions,” it read.
“We hope to resolve this matter and will not be providing any further comment at this time.”
It leaves Arsenal with few options at the back, although they are close to completing a deal for William Saliba.
However, he will be loaned back to Saint-Etienne for the 2019-20 campaign.