Kaka recently expressed interest in a future move to MLS and on Friday Tim Leiweke, president of Galaxy's owners AEG said they wanted to sign the 2007 World Player of the Year.
"We're well aware of Kaka's interest in MLS, and we in turn have made it very clear to him that he's aware of our interest in him," said Leiweke.
"We have a great relationship with Real Madrid, and just as we worked through a player with them six years ago (Beckham), I'm absolutely convinced we could find the right deal this time, too.”
In an earlier interview with the Galaxy's official podcast Leiweke said they wanted a player to boost the club's profile in the Hispanic community and said Beckham, who played with Kaka at AC Milan, had taken a leading role in the bid.
"The guy who introduced us to this player is David Beckham. David has done the yeoman's work on our behalf at introducing us, getting him comfortable, getting his family comfortable, talking about L.A. and the organisation," Leiweke said.
Beckham will play his final competitive game with the Galaxy in Saturday's MLS Cup final against Houston Dynamo.
The 30-year-old Kaka has long been linked with an eventual move to MLS and his comments after the friendly in New Jersey earlier this month between Brazil and Colombia, indicating his desire for a move in the future, intensified speculation.
Galaxy's Ireland striker Robbie Keane said he would be delighted to see the Brazilian take Beckham's spot.
"We want as a team and an organisation to push on and be the best that we can. To attract players like Kaka, he will have big shoes to fill with David going," said Keane.
"We welcome any top players, Kaka was voted one of the best players in the world only a few years ago. We'd welcome anyone with his stature and ability to the Galaxy but ultimately it is down to Tim Leiweke and (coach) Bruce Arena," he said.
Beckham's involvement in the attempted deal for his replacement is another fascinating aspect of the former England midfielder's future plans.
The 37-year-old has said he intends to have one more playing challenge after leaving Galaxy but will then return to the US to take an ownership role although he has not indicated where that involvement would be or what form it would take.
Beckham has an option to take an ownership share in a future expansion team in MLS but Leiweke suggested his involvement will be with the Galaxy.
"David is going to continue to be a part of this organization going forward because there is no one better to go out and get some of the best players in the world to come here than David," he said.
Major League Soccer commissioner Don Garber also noted that the Galaxy were keen to keep Beckham involved at their club.
"I have heard that Tim Leiweke said that he would be interested to talk to David about how he might be involved in the Galaxy, this is all very real-time," he said.
At his news conference on Thursday, Beckham mentioned a future role, in some form, as a recruiter of talent for MLS.
"Just because I'm not playing here anymore after the weekend, my commitment stays the same. And I will do anything to keep these players coming over like Robbie Keane, like Thierry Henry, and anything I can do on that side of things I want to do," he said.