Martin and his manager Ralph Carr met the Giants on Thursday, three days after Richmond announced that they had withdrawn their current offer to the 22-year-old.
"We took up the invitation to meet with Martin and his manager Ralph Carr and showed them around our new training facility and Skoda Stadium but we have made a decision not to pursue further discussions," GWS general manager of football Graeme Allan said.
"We will continue to explore a number of opportunities with other players with a view to add more experience to our playing list and we wish Dustin Martin all the best for his career ahead."
Earlier on Friday, Richmond announced they had signed another of their best young midfielders, Reece Conca, to a three-year deal.
Conca, 21, produced some eye-catching performances in his 17 games this season as the Tigers reached the finals for the first time in 12 years.
Conca was taken by the Tigers at No.6 in the 2010 national draft.
The West Australian had been linked with a potential move to Fremantle or West Coast before signing the new contact that will keep him at Punt Rd until at least the end of the 2016 season.
"It was never in doubt that I would re-sign with Richmond. It was just a matter of sorting through the details with the club and my manager," Conca said.