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Former England captain David Beckham is leaving Los Angeles Galaxy next month after six years at the American Major League Soccer (MLS) club.
He intends to quit after the season-ending MLS Cup final on 1 December.
In a statement Beckham, 37, said: "I wanted to experience one last challenge before the end of my playing career."
Melbourne Heart claim they are in talks with Beckham over a "10-game stint" in Australia, despite his camp saying he had "no plans" to play in the A League.
Heart chief executive Scott Munn told BBC Sport: "We are incredibly serious and working tirelessly with David's representatives to table a compelling offer which is commensurate with the international player that David is.
"We're talking to David about a 10-game stint. Football is the third code behind Australian Rules football and rugby league. We think David takes it to the next level."
Perth Glory are another Australian A League club interested in signing Beckham.
"He would be an asset for any club, but more importantly, we would love to have him here," said chief executive Paul Kelly.
Beckham joined Manchester United as a 14-year-old and went on to make 398 appearances for the club, winning six Premier League titles and the Champions League.
He moved to Real Madrid in a £25m deal in 2003, winning the La Liga title in 2007 just weeks before his switch to the United States.
The midfielder made his England debut against Moldova in 1996 and captained the side from 2000 to 2006, making his 115th and final England appearance in a 3-0 win over Belarus in 2009.
Beckham insisted his links with MLS would continue should he join another league.
"I don't see this as the end of my relationship with the league as my ambition is to be part of the ownership structure in the future," he said.
Beckham's initial contract with the league included a clause allowing him to be involved in owning an MLS expansion team down the line.
MLS commissioner Don Garber praised Beckham's impact on MLS.
"There is no doubt that MLS is far more popular and important here and abroad than it was when he arrived," Garber said in a statement.
"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."
However, American football journalist Sean Wheelock, who is a regular guest on the BBC's World Football phone-in radio show, believes Beckham has not had the same impact on the game in the United States as Brazilian superstar Pele did in the 1970s when he played for New York Cosmos.
"The league has grown exponentially but it would have grown anyway," he told BBC Radio 5 live. "I see a direct link with players like Thierry Henry and Robbie Keane coming over to MLS but I'm not going to buy into the argument that he [Beckham] revolutionised it because the league has been making slow, gradual climbs forward."
His stint in Los Angeles brought success in the form of the Galaxy's third MLS Cup crown last year.
Twice he moved on loan to AC Milan during the American off-season. The second time, he tore an Achilles tendon that kept him out of a large part of the following MLS season.
Beckham was voted the league's best player by supporters last season.
Former Galaxy team-mate Chris Birchall, now playing for Columbus Crew, says Beckham will still have a host of clubs interested in signing him.
"His speed has gone but that's about it," said Birchall, 28. "His technical ability, his passing and his free-kicks are second to none."