Beckham’s contract with the Galaxy expired at the end of 2011 and he was expected to join Paris Saint-Germain in January, but the French side revealed that a deal for the former England captain is off because he does not want to uproot his family.
The 36-year-old is now in negotiations to re-sign for the MLS club but it appears that a new contract is not close to being agreed.
"We're down the road," Leiweke told The LA Times.
"I am confident we'll get it done. We have work to do, but we're not there yet. And as much as I didn't get too low when they [PSG] were planning the parade down the Champs-Elysees … now as much as the media is reporting that we're done, I've been on the phone with [Beckham's] people all day.
"Trust me, we're not done. I wish we were done.
"[Galaxy coach Bruce Arena] has given him the ability to go [to the Olympics].
"We are being supporters and fans of the Olympic Games in London. We helped put up seed money for their bid and [AEG's] O2 Arena in London is hosting gymnastics and basketball. We could not be bigger supporters of the Olympics."
Beckham spent five seasons in LA having signed from Real Madrid and impressed in his final year as he led the side to the MLS title with the second most assists in the league and picked up the comeback of the year award.
His global appeal and ability means he has had a number of lucrative offers to move elsewhere in the world, but Leiweke is still confident that he will decide to remain in LA in the future.
“It's not just Paris. He's had other offers," he said.
"And every one is greater and more expansive, bigger than anything we're talking about. In addition to that, we have the added complication that David has the right to purchase a franchise in this league.
"There are a lot of issues we are going through. We are committed to David and he has always felt strongly about the Galaxy in LA. and it does not surprise me that his family would prefer to stay in LA.
"I believe we're heading in the right direction. I'll repeat what I said in November: I believe David Beckham will play for the LA Galaxy this year."
Le Parisien, meanwhile, reports that Beckham did not want to move to the leaders in France’s Ligue 1 due to it being too physically demanding for him.
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