Spagnuolo is scheduled to interview with the Saints on Wednesday, according to a league source.
Payton said Tuesday it was apparent before the season ended that Williams, with his contract expiring, likely would join Fisher, an old friend under whom he coached in Tennessee for a decade.
"It was a matter of getting through the season and not allowing it to be distraction," Payton said, adding that the Saints and Williams never discussed an extension.
Spagnuolo was defensive coordinator for the New York Giants for two seasons (2007-08) before coaching the Rams for the last three seasons, compiling a 10-38 record. He has an extensive defensive background, coaching on that side of the ball for eight seasons in Philadelphia under esteemed coordinator Jim Johnson.
Follow Jason La Canfora on Twitter @JasonLaCanfora
The Associated Press contributed to this report.