"Negotiations are getting more intense. I hope to manage to arrange for him to stay here at least until June 30 on loan," Galliani told the club's in-house TV channel.
"Then he could return to Los Angeles and come back in January. It's a complicated path, as we have always known, but I'm trying."
The clubs are close to an agreement for the loan extension which should be finalised by the end of the week, the Los Angeles Times reported on its website.
Beckham is due to return to Galaxy on March 9 if a deal is not reached. The clubs had been in talks over a permanent transfer but were unable to agree on a fee.
The 33-year-old midfielder initially joined Milan to stay in contention for England selection during the U.S. close season but enjoyed playing top class football again so much he set his sights on a long-term stay.
Beckham has settled well, scoring two goals in nine Serie A games, although the level of his recent performances has dipped as Milan have run into some poor form.
They were knocked out of the UEFA Cup by Werder Bremen last Thursday and are 12 points behind league leaders Inter Milan.
‘After Manchester City’s recent form, maybe they’re the underdogs against Manchester United!’ Former Red Devils defender on this weekend’s derby
‘Arteta, Alonso, Emery, me… none of us were physical players – we needed the understanding of the game. That probably helped us move into management’: Premier League boss reveals reasons for natural career progression
