The deal is for five years, league sources told ESPN Adrian Wojnarowski.
Williams is the Suns' seventh coach in an eight-season span.
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.
Williams left the Oklahoma City Thunder as associate head coach in the aftermath of his wife's death.