West Ham owner David Gold believes it would be a "fantastic statement" if West Ham could sign Beckham.
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.
David Gold: "He was an ambassador for the London 2012 bid and it would be a fantastic statement if he came to us.
"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."
Beckham says he has yet to make a decision on his future.