Mare spent last season with New Orleans after a 10-year stint with the Miami Dolphins.
In his career, the Syracuse product has made 255 of 320 field-goal attempts and 347 of 352 extra points.
He was chosen for the Pro Bowl in 1999 when he established a then-NFL season record by converting 39 field-goal attempts.
In his only season in New Orleans, Mare, 34, made 10 of 17 field goals and added all 34 extra-point attempts.
"I know the goal for them was for me not to leave here," Mare told the Rocky Mountain News after his visit with the Broncos this week. "But my goal is not to leave here. "(The Seahawks have) been really positive and interested and we'll see how it goes."
Brown spent five seasons with the Seahawks before signing as a free agent with St. Louis.