Former American International Group Inc. Chief Executive Maurice R. "Hank" Greenberg will appeal not just the parts of a federal court ruling that didn't go in his favor, but is seeking to revive claims that had been dismissed in 2013, his legal team said in a filing Thursday.

The notice of appeal comes three days after the U.S. Court of Federal Claims ruled in favor of Mr. Greenberg and other shareholders in the class-action suit that the government had violated the law in demanding a 79.9% equity stake in AIG in its 2008 bailout of the financial-services conglomerate.

But the judge said he awarded none of the $40 billion being sought because he had to take into consideration that AIG's alternative to the government's harsh deal terms was to file for bankruptcy, an outcome that likely would have left shareholders with nothing.

Mr. Greenberg's legal team, led by David Boies, said Tuesday that his team would appeal the matter of the damages because not doing so "would allow the government to act illegally with impunity and without remedy or consequences."

The surprise part of Thursday's notice was that Mr. Greenberg wants the appellate court to review a 2013 decision by the same judge, Thomas C. Wheeler, in which he dismissed "backdoor bailout" claims, as Mr. Greenberg's lawyers have called them in court filings. Those involve allegations that the government unlawfully gave billions of dollars to AIG's counterparties on Wall Street and elsewhere to make them whole.

Judge Wheeler had dismissed those high-profile allegations after lengthy debate by AIG's board about whether it would participate in the lawsuit. The board voted not to participate after a "rigorous review process and reached a reasonable decision, well explained in filings with the court," Judge Wheeler wrote at the time. He said Delaware law required the court to give deference to AIG's board, and so he dismissed the claims that AIG had been harmed by the government when it repaid AIG's counterparties.

A spokeswoman for the Justice Department said lawyers there "are reviewing the court's decision" from Monday and had no comment on Mr. Greenberg's effort to reopen the dismissed claims.

Legal experts note that the government also might opt to appeal, and the case ultimately could make its way to the Supreme Court.

If successful in reviving the claims, the government and Mr. Greenberg's legal team could find themselves back in a courtroom, slugging it out on this additional front, the legal experts said. Monday's ruling followed a 37-day trial last fall.

Thursday's filing renewed concerns among some current AIG shareholders that the company itself potentially will be on the hook for damages. The concern springs from a provision in AIG's bailout agreement that obligates the company to compensate the U.S. for costs arising from litigation over the pact.

Some investors have worried about the provision since the suit was filed in 2011, but the matter seemed largely ended Monday when Judge Wheeler didn't award damages. AIG's shares jumped after news of his ruling. They dropped on news of the notice of appeal Thursday.

AIG said in a financial filing earlier this year that it would defend against any effort by the government to seek to collect from it. Some lawyers who have followed the lawsuit believe the company would have strong defenses.

The AIG bailout has been controversial over the years for many reasons, including the alleged "backdoor bailouts." In short, these entail a decision by the government to purchase tens of billions of dollars of mortgage bonds from AIG counterparties, which include big names on Wall Street and in the international banking world, to make those parties whole. AIG's liquidity problems during the crisis stemmed largely from collateral calls tied to the insurance it had sold to protect the Wall Street firms and banks against losses on these bonds.

In his Monday decision, Judge Wheeler wrote that "the government was able to minimize the ripple effect of an AIG failure by using AIG's assets to make sure the counterparties were paid in full on these transactions." AIG's counterparties "also received complete releases from AIG for all legal action, including any potential fraud or misrepresentation claims," the judge noted.

AIG's board voted unanimously in January 2013 not to join Mr. Greenberg's lawsuit, partly to avoid reputational harm that they feared could result from signing on, people familiar with the matter have said.

Write to Leslie Scism at leslie.scism@wsj.com

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