Leeds are the club he has chosen, though, with manager Neil Warnock offering the former Liverpool man a two-and-a-half-year deal.
"He is a former international who, for whatever reason, has been allowed to leave Aston Villa," Neil Warnock said.
"He is a player I have tried to sign a few times before and I'm pleased he turned down West Ham to come here."
He gave a nod to the difficulty of his circumstances at Villa Park, but stopped short of criticising Lambert.
"The situation I was in wasn't great. I've not been in and around the first team - I've been with the reserves and I don't want to criticise," he said.
On snubbing West Ham, he added: "I was on my way there but Leeds made an effort to sign me and make me feel wanted."