The FFA claims Beckham's management team has approached them about moving Down Under for a 10-game stint following the conclusion of the MLS season in America in two weeks' time.
An FFA spokesman said: "The approach from David Beckham's people is another sign of how much the Hyundai A-League has grown in stature on a global basis.
"Beckham is a superstar on the world stage and he would be another massive signing for the A-League after the arrivals of Alessandro Del Piero, Emile Heskey and Shinji Ono. But this approach remains at a very preliminary stage."
Sydney or Melbourne are considered the two most likely destinations, although for the move to happen FFA must support the deal financially.
The former Manchester United man is likely to demand a large fee, out of the reach of most Australian clubs.
That would mean Beckham's pay could come directly from the pockets of FFA chief Frank Lowy, the co-founder and chairman of shopping centre giant Westfield who has a fortune estimated at around US$5b.
Italian superstar Del Piero is on a two-year contract reportedly worth £1.25m a season at Sydney FC.
Heskey, one of Beckham's former England team-mates, is also playing in Australia with the Newcastle Jets.
Melbourne Victory coach Ange Postecoglou ruled out a move for the former England captain but other clubs showed more interest.
"That's great if he is (interested in coming to the A-League). Would we be interested? No," Postecoglou told foxsports.com.au.
Perth Glory chief Paul Kelly was unaware of any approach from Beckham's representatives but is happy to talk.
"I don't know anything about it. I know he's being touted around, but there's been no conversation," Kelly said. "As usual, I'll always say I'm happy to speak to anyone."