UPDATE 1-Greenberg won't sue AIG over US government claim-lawyer

* Boies tells CNBC that Greenberg moving forward

* AIG board decided Wednesday not to join lawsuit

Jan 10 (Reuters) - Hank Greenberg will not sue AIG over the company's decision not to join his lawsuit against the U.S. government, his lawyer said on Thursday.

Greenberg, who was chief executive officer of American International Group Inc for decades, has sued the government over the insurer's 2008 rescue, alleging it was done on unfair terms that harmed shareholders. He is seeking billions of dollars in damages and had demanded that AIG join in the case as well.

The company's board, amid a fevered public backlash, said Wednesday that it would not join the lawsuit and would not let Greenberg pursue claims on its behalf. That decision created the possibility that Greenberg could sue AIG for interfering with his case, an action his lawyer said he would not take.

"We're not going to be doing that," David Boies told CNBC in an interview. "We're not looking to sue AIG."

An AIG spokesman declined to comment.

That AIG's board had even considered joining Greenberg's lawsuit caused a huge public uproar this week, with thousands of people taking to social media and accusing the company of ingratitude or worse.

One group of congressmen sent a letter to AIG's chairman in which they advised: "Don't do it. Don't even think about it."