The former France youth international was widely regarded as one of the division's best young performers last term as, in an abridged campaign curtailed by the Covid-19 pandemic, he helped the club to a sixth-place finish.
His move to Monaco represents something of a coup for the principality outfit, who unlike Reims will not compete in Europe next term after they finished outside the qualification places for both the Champions League and Europa League.
Wolves, Southampton and Roma were among the other clubs chasing the ex-Paris FC man's services, but Disasi has instead bought into Niko Kovac's vision for the club and put pen to paper on a long-term deal on the south coast.
"I'm delighted to be involved with AS Monaco," the 22-year-old told the club's official website. "This is a great opportunity for me to demonstrate my qualities at the highest level and to continue my progress at a club with high ambitions.
"I'm looking forward to starting training with my new team-mates and I'm going to put all my energy into helping the team achieve its goals at the end of the season."
Sports director Paul Mitchell expressed his elation at securing Disasi's talents too, adding: "We are delighted that Axel has chosen to join AS Monaco.
"His qualities as a defender, his maturity and his experience in the French league, despite his young age, will be additional assets that we are happy to make available to Niko Kovac's squad for the next five seasons."
Arsenal previously contacted Reims in April as the Gunners weighed up a defensive revamp under Mikel Arteta, with Disasi seen as a potential long-term talent to be nurtured at centre-back.
The club have welcomed back one young French talent, however, already this off-season, with William Saliba returning from his loan spell at Saint-Etienne.
Elsewhere, Arsenal are waiting on Pablo Mari's return to fitness after the Spaniard underwent ankle surgery, while Konstantinos Mavropanos has been loaned out to Stuttgart for the coming season.