The deal is for five years, league sources told ESPN's Adrian Wojnarowski.
Williams, 47, is the Suns' seventh coach in an eight-season span. The team fired Igor Kokoskov last month after one season.
"Monty brings a wealth of NBA experience, both as a coach and former player, in addition to being a high-character individual who will infuse basketball wisdom and life lessons into our locker room," Suns general manager James Jones said in a release announcing the move. "Monty is well respected for his coaching pedigree, leadership and commitment to the community, all of which make him the ideal person to lead our team moving forward."
Williams returned to coaching this season as Brett Brown's top assistant. He spent the previous two years in the San Antonio Spurs' front office following the death of his wife, Ingrid, in early 2016.
"Anybody who pays attention even a little bit understands he's elite people," 76ers coach Brett Brown said. "He's emerging, relatively young coach who's incredibly relational and has been studying how to be an NBA coach. He's been around and I hope he's taken something from our program."
Williams left the Oklahoma City Thunder as associate head coach in the aftermath of his wife's death. He and his children relocated to San Antonio, where Williams had extended family.
Before his time with the Thunder, Williams coached New Orleans for five seasons. He reached the playoffs in his first season (2010-11) and his last (2014-15).
Since 2012-13, the Suns are 199-375, tied with the Orlando Magic for the worst record in the NBA in that span.