Beckham announced on Tuesday that he would be leaving Los Angeles Galaxy at the end of the MLS season, and it is expected that he will now receive offers for his signature from clubs all around the world.
The former England captain revealed last week that he had no plans to move to Australia, but that has not prevented Melbourne Heart making their interest known just hours after Beckham announced his imminent departure.
CEO Scott Munn told reporters: "What I can say is that it is certainly real. We are putting forward an offer. It is compelling and the opportunity is here for him to come here.
"Let's let David get through next week, play in the final of the MLS, and then I am sure that he will assess every offer."
The 37-year-old has scored 20 goals in 116 matches for LA Galaxy.