Martin has two years to run on his current deal, but his love for the harbour city could pave a way for him to join the Swans.
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Martin lived with his father in Camden and has a strong affiliation to the Sydney lifestyle, which could draw him away from Tigerland.
“Dusty loves Sydney,” Carr told The Herald Sun’s Phil Rothfield.
“He’s there a lot and he originally lived in Camden with his dad. You’ve got to live there to really know how good it is.
“He loves rugby league, too, and he was hanging out with Cameron Munster the other day.”
Carr also revealed Martin wants to be Australia’s version of Tom Brady and play into his late thirties.
With Lance Franklin in the latter stages of his career, the Swans are in need of a new marquee player to entice fans back to the SCG.
Martin is arguably the most recognisable footballer in Australia and his draw power would be double that of any player currently at the Swans.