Vitt had served as the saints' assistant head coach since 2006.
Payton, 53, told WWL Radio Monday that he "definitely" planned on being back with the Saints. He has four years left on his current deal.
The New Orleans Times-Picayune reported Monday that Saints general manager Mickey Loomis "would be open to trading" Payton. Payton's current five-year contract extension is worth $45 million and runs through 2020.