Beckham will play for Los Angeles Galaxy for the last time in the Major League Soccer Cup final against Houston Dynamo on 2 December (AEDT).
Heart chief executive Scott Munn said while the 37-year-old English star would be fielding offers from around the world and other A-League clubs, Heart believes it is in with a chance.
"I can't put a percentage term on it," Munn said. "What I can say is that it's certainly real, we're putting forward an offer, it's compelling and the opportunity is here for him to come here.
"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."
Heart's offer will be for a 10-week guest player stint, including finals, believed to be worth between $1-2 million.
Munn said while Beckham would provide a huge off-field boost, his form and fitness meant he could immediately also be a significant on-field asset.
Heart skipper Fred, who played against Beckham in the MLS during his time at DC United and Philadelphia Union, said Beckham would have a huge effect on the club and league if he came.
"Oh my gosh. Every game is almost full house," Fred said of Beckham's drawing power in the US.
"Especially I think the women come to watch him.
"But he's good. If he comes it will help the league 100 per cent."
Fred said Beckham would also help his teammates.
"He's a good player, he's still fit, I think (it would be) good for us to get experience with him," he said.
"It would be amazing. Everybody knows about him."
Meanwhile, the man who brought Dwight Yorke and Alessandro Del Piero to Australia said that the A-League should do everything within its power to sign up Beckham.
Respected player agent Lou Sticca, whose strong rapport with Galaxy paved the way for the MLS club to play in Australia three times in the last five years, said the A-League should not miss this once-in-a lifetime opportunity.
"I am a traditionalist and I don't usually like publicity stunts," Sticca said.
"I'm the sort of bloke who would like nothing more than go to Sunday arvo matches in the middle of winter with a scarf wrapped around my neck.
"But the game has moved on and we play in summer.
"With Beckham everything is different.
"He is special and the normal rules do not apply with him.
"Australian football needs to get out of its comfort zone and invest in a player like Beckham.
"Ok, it would be only a guest stint but look at the massive exposure he would give the league.
"You can't buy that sort of stuff.
'I am sure that Beckham's management will be fielding offers from numerous destinations in Europe, Asia, the Middle East and other parts of America.
"I have forwarded to Beckham's management a number of offers from A-League clubs and I hope they will receive a favourable consideration."
Sticca said the league's standard was improving every year but he said it is still not where it wants to be.
"It is still fighting for air in a very competitive and saturated sports market and a guy like Beckham would give it enough oxygen that will last well beyond his departure," he said.
"We have seen how the arrival of Del Piero, Emile Heskey and Shinji Ono has boosted the A-League in the last three months.
"Beckham would provide a cut-through to non-traditional football fans and this can only help the A-League broaden its appeal.
"This is a unique opportunity for Australia to tap into the world market and no expense should be spared."
Additional reporting by Philip Micallef