The 37-year-old is expected to end his six-year stay in Major League Soccer on December 1 after the MLS Cup clash against Houston, with Melbourne Heart and Western Sydney Wanderers emerging as possible contenders for the former Manchester United and Real Madrid star’s signature.
"We're putting forward an offer. It's compelling and the opportunity is here for him to come here," Melbourne Heart chief executive Scott Munn told reporters.
"What I can say is that it's certainly real... That is absolutely legitimate.
"Let's let David get through next week, play the final of the MLS and hopefully he'll have a win. Then I'm sure he'll assess every offer."
Beckham’s management company have denied reports linking their client with a move to Australia but Perth, Adelaide and the Gosford-based Central Coast Mariners all also voiced interest.
Meanwhile Sydney FC and Melbourne Victory have indicated that they will not pursue the player.
The 115-capped international will join the likes of marquee signings Alessandro Del Piero, Emile Heskey and Japan's Shinji Ono if he is lured to the A-League.