The Vail, Colo., firefighter was invited last week to the Saints' rookie minicamp, the New York Post reported Monday.
Sutter was an all-conference defensive back when he attended the University of Colorado. A fifth-round pick in the 1998 draft, Sutter signed with the Carolina Panthers, but dislocated his shoulder in his first game to end his NFL career.
Sutter later played a season in Europe for the Barcelona Dragons.
Some 17 million viewers watched Sutter marry Rehn in December 2003 during the first season of "The Bachelorette."