Having turned down a new contract, Beckham has vacated one of the club's designated player slots - set aside for higher earners - and Galaxy have repeatedly been linked with a move for Lampard as they seek a high-profile acquisition.
The Chelsea midfielder is out of contract in the summer and the club have shown no interest in offering a new deal to a player who is only eight goals shy of Bobby Tambling's record of 202 goals in the club's colours.
Now Arena says Lampard is a player of interest, as Galaxy look to make an announcement soon.
Arena said: "I wouldn't be surprised if over the next week or so we're going to have some sort of decision with a designated player.
"We wouldn't rule a move for Lampard out. It's always a possibility."