Glory owner Tony Sage had already revealed the club had "put its hat in the ring" to sign the former Manchester United and Real Madrid star for a 10-game guest stint.
Sage has been in contact with leading agent Lou Sticca, who is involved in the deal to sign Beckham and was also responsible for helping to bring Alessandro Del Piero to Sydney FC and Shinji Ono to Western Sydney.
Western Sydney Wanderers and Central Coast Mariners are also keen to sign Beckham.
Millionaire ad man John Singleton has indicated he would back the Mariners' bid.
"There's stuff going on behind the scenes. We're definitely putting our hand up," said a Mariners spokesman.
"He would fill the stadium and change the game up here."
In his six seasons with the Galaxy since signing in January 2007, Beckham has appeared in 98 championship games and his stint coincided with a period of unrivalled growth for both the club and the MLS.
His ability to further boost the profile of the A-League - on the back of the success of off-season foreign signings Del Piero, Ono and Emile Heskey - will appeal to new FFA boss David Gallop.
Seven MLS expansion teams have debuted since 2007 while 15 of the league's 19 teams now play in soccer-specific stadiums - an increase of five before 2007.
Attendance at those stadiums are at unprecedented highs with an average of 18,807 fans at games during the 2012 MLS regular season.
Exposure and recognition of the Galaxy and MLS are at all-time highs with the league maintaining lucrative television broadcast deals while the Galaxy reached a 10-year broadcast arrangement which is the most lucrative in MLS history.
- AAP