The 37-year-old Beckham has announced that his six-season stint in U.S. Major League Soccer will end with the Dec. 1 MLS Cup against Houston.
Five Australian clubs have confirmed interest in bringing Beckham to the A-League, with Melbourne Heart and Western Sydney Wanderers considered the top contenders by the domestic media due to the cosmopolitan nature of the two cities.
Last week, Beckham's management company denied reports he was headed to Australia.
Perth, Adelaide and the Gosford-based Central Coast Mariners all expressed interest, while Sydney FC and Melbourne Victory have previously indicated they won't pursue Beckham.
"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 said Tuesday
"What I can say is that it's certainly real ... That is absolutely legitimate," Munn added. "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 has been a big influence on the growth in popularity of the MLS since he joined the Galaxy, and even a short 10-week stint in Australia could generate extra interest the fledgling A-League.
The Australian competition is already breaking its own records this season after the recruitment of Alessandro Del Piero, Emile Heskey and Japan's Shinji Ono.
Beckham hadn't given any overt indications he was planning to leave the Galaxy after this season with a year left on his deal, and last week his management pointedly denied rumours linking him to a short-term stint in Australia.
But he released a statement in conjunction with the Galaxy late Monday saying he had an "incredibly special time" playing for the club, "However, I wanted to experience one last challenge before the end of my playing career."
The former Manchester United and Real Madrid star and his wife, former Spice Girl Victoria, have been living in Los Angeles with their sons Brooklyn, Romeo and Cruz.
"There is no doubt that MLS is far more popular and important here and abroad than it was when he arrived," MLS Commissioner Don Garber said. "David has achieved great things on and off the field during his time with the Galaxy, and he will always be an important part of our history."