The 37-year-old LA Galaxy midfielder remains one of the game's biggest names and his presence along with Italian superstar Alessandro Del Piero would help ratchet up the competition's global image.
FFA said Friday Beckham's management had offered his services for the current season. Reports said he wanted to play a 10-game guest stint once the American Major League Soccer season ends in a fortnight.
"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," said an FFA spokesman.
"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."
The FFA said it was exploring all options available.
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 multi-million dollar fee which would be out of reach of must Australian clubs.
The Sydney Morning Herald speculated that the newly-formed Western Sydney Wanderers would be the most realistic option with FFA owning the team.
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$5 billion.
Lowy previously offered to put the money up if the Wanderers wanted to sign German international Michael Ballack. But the deal fell through when coach Tony Popovic said he was not interested, the Herald said.
The team signed Japanese international Ono instead.
An indication of the sort of money Beckham could demand is reflected in Italian superstar Del Piero being on a two-year contract reportedly worth US$2 million a season at Sydney FC.
Heskey, one of Beckham's former England teammates, is also playing in Australia -- with the Newcastle Jets.
During his time with the Galaxy, Beckham has played against Sydney FC, Newcastle and Melbourne Victory in friendly matches.