The England international tore his Achilles tendon in March and missed the World Cup in South Africa after undergoing surgery.
"I believe there's a chance he can begin training by the end of August," Arena told reporters following the MLS All-Star game against Beckham's first club Manchester United on Wednesday.
The MLS regular season finishes at the end of October but the Galaxy, who lead the Western Conference, are expected to be involved in the November play-offs that decide the title.
It was too early to say whether the 35-year-old could be involved in the playoffs, Arena added.
"At this point in time there's no way to predict whether he'll be able to play this year or not.
"These are difficult injuries ... for any athlete in any sport, at this point in time we can't predict whether he is going to be able to play or not," he said.
"There's a possibility he could play towards the end of the MLS regular season, but at this point in time it would be difficult to say yes or no," he added.