Green, 32, will be out of contract at the end of June and has not agreed new terms despite the Hammers' promotion back to the Barclays Premier League.
Former captain Matthew Upson left Upton Park as a free agent last year and Green is now set to follow suit having been linked with Tottenham, QPR, West Brom and Spanish side Malaga .
Gold wrote on Twitter: "Robert Green is a free agent he has chosen to move on. Matthew Upson did the same thing. "Nobody wants Rob to leave but nothing is for ever, he must do what is best for him and we must do what is best for us."
Green was an unused substitute as England beat Belgium 1-0 at Wembley yesterday in their final warm-up ahead of Euro 2012.
He joined the Hammers from Norwich for £2million in 2006 and made 241 appearances. Manager Sam Allardyce has lined up Bolton's Jussi Jaaskelainen as Green's replacement.
Press Association Sport understands the 37-year-old, who is also out of contract, will sign next week subject to passing a medical.
Jaaskelainen will compete with Stephen Henderson, whose loan move from Portsmouth became permanent last month, to be the club's number one next season.