Buford will become CEO of Spurs Sports & Entertainment.
Jabari Young of The Athletic first reported the news Friday.
Buford played college basketball with the Texas A&M Aggies and Oklahoma State Cowboys before joining the Spurs coaching staff in 1988. He left in 1992 for one-year stints with the Los Angeles Clippers and Florida Gators before returning to San Antonio in 1994.
The 58-year-old executive was promoted to general manager in 2002 and has held the position since. He's been with the organization for all five of its NBA championships, the most recent coming in 2013-14, and he was named NBA Executive of the Year in 2013-14 and 2015-16.
"The best way to put it would be to say we'd be lost without him," Spurs coach Gregg Popovich told reporters in 2016. "I think those few words say it all. His organizational abilities, his foresight, his ability to plan ahead of time and make judicious and wise decisions is off the charts. We would have had a hard time keeping this together for this long if he wasn't here."
He'll take over a roster without the three pillars of the Spurs' dynasty: Tim Duncan, Tony Parker and Manu Ginobili.
Though the franchise still features star power in LaMarcus Aldridge and DeMar DeRozan, it has struggled to keep pace in an era dominated by the Golden State Warriors as elite players have chosen to team up with each other on select squads.
Numerous high-profile players changed organizations over the past month via trades or free agency, and none of them landed in San Antonio. DeMarre Carroll was the team's top addition.
Wright will be tasked with getting the Spurs back into championship contention after taking over for one of the most decorated general managers in NBA history.