Beckham has confirmed he is leaving Los Angeles Galaxy after next month's MLS Cup final and the 37-year-old has been attracting interest from clubs across the world.
West Ham have been linked with the former Manchester United star in the past and co-chairman Gold would be keen to offer Beckham a return to the Premier League.
"He was an ambassador for the London 2012 bid and it would be a fantastic statement if he came to us," said Gold.
"He's an East End boy and it would be a fitting end to his career.
"We've also been mindful of his value as an ambassador to help us secure the Olympic Stadium. "
Melbourne Heart are one of several Australian clubs to show an interest in Beckham, while Paris St Germain are also considering a move for the ex-England captain.
However, Beckham says he has yet to make a decision on his future and simply felt it was the right time to announce he would be leaving the Galaxy.
In an interview on the Galaxy's YouTube channel, he said: "What's next? I haven't decided yet.
"I'd been thinking about this for a few months and I came to the decision a couple of weeks ago. We were just about to head into the play-offs so I felt it wasn't the right time to announce it.
"Now we're through the play-offs and in the MLS Cup final which is going to be in our stadium and in front of our fans I felt it was the right time to do so."