Beckham will play for the LA Galaxy for the last time in the US Major League final on December 1.
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, the Heart believed they were in with a chance.
"I can't put a percentage term on it," Munn told reporters on Tuesday.
"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."
The Heart's offer will be based on a 10-week stint including finals and be worth between $1-2 million.
He said while Beckham would obviously 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, said his effect on the club and the league would be enormous 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."
Fred will return from a three-game absence with a hamstring injury for the Heart against Newcastle at AAMI Park on Saturday.