The Australian ironwoman, who caused controversy after being photographed in a hotel toilet with sportsman Sonny Bill Williams in 2008, apparently started seeing Burgess two months ago.
A friend told The Daily Telegraph: "She's moved back to Sydney from Perth to be with him, so that shows you it's serious.
"They struck up a friendship and soon began dating and now they train together, hang out and go out together."
The couple, who reportedly met at Pyrmont's Majestic Massage Clinic, were spotted having breakfast in Sydney together last weekend.
The friend added that Burgess is "cool" about Falzon's toilet scandal.