Insurer American International Group Inc's once-desperate financial state has started to stabilize, a government agency said on Monday, as an influential lawmaker said he would look at easing the terms of the insurer's federal bailout once more.

AIG's shares shot up, rising more than 16 percent in afternoon trade to $46.44.

AIG, the recipient of billions of dollars in government support, started to show signs of stabilizing in mid-2009, as financial markets improved, congressional investigators said in a report on the status of the government's assistance to the company.

The insurer was rescued by the government in September 2008, after losing bets that it made on the U.S. housing market threatened to drive the firm into bankruptcy.

Rep. Edolphus Towns, a Democrat who chairs the Government Reform Committee, told staff to take a look at a proposal from former AIG Chief Executive Officer Hank Greenberg that could make it easier for the insurer to repay federal obligations, a committee spokeswoman told Reuters on Monday.

The proposal would cut the government stake in AIG from the current 80 percent and trim interest rates on AIG's government loan. It also would extend the term of the loan, giving the company more time to repay, the spokeswoman said.

However, it is still unclear whether U.S. taxpayers would ever be repaid fully, or whether AIG could pull off the restructuring of its business, according to the report compiled by the Government Accountability Office (GAO), the investigative arm of Congress.

The GAO said the U.S. government remains exposed to risks, including credit risk and investment risk, which could result in the Federal Reserve and Treasury not being repaid in full. (The GAO's full report is available at http://www.gao.gov/new.items/d09975.pdf)

Towns' spokeswoman said the congressman has not yet spoken with the U.S. Treasury Department or the Federal Reserve about Greenberg's proposal.

A NEW START?

AIG's total bailout package, already rewritten three times since September 2008, has swelled to as much as $182.5 billion. The rescue includes the government's purchase of toxic assets to relieve the insurer of billions of dollars in liabilities, and a funding facility that the insurer has yet to draw in full.

AIG currently owes about $80 billion in federal loans.

To repay its debts, AIG has tried to sell off assets; but so far it has raised net proceeds of only $4 billion, hampered by tight credit markets and buyers looking for bargains.

The possibility of a fourth revision to AIG's bailout appeared to spark the rally in AIG's stock on Monday, said Jon Najarian, co-founder of optionmaster.com.

He said the bullish sentiment appeared to be driven by retail investors and day traders, with triple the normal volume taking bets that the shares could trade as high as $65 by next month.

Najarian said there were no signs that big institutions were involved in the surge in activity in the shares on Monday.

AIG's shares have rallied strongly since the insurer installed Robert Benmosche to be its new CEO, rising from $28.37 on August 10, the day he took the helm, to as high as $55.90 in late August.

Benmosche, AIG's fourth CEO since June 2008, has said he expects the company will be able to repay its federal debts, and hopes there will be something left over for shareholders.

Former CEO Greenberg, who ran AIG for 38 years, last month said he had agreed to help Benmosche, giving him advice on how to turn around what was once the world's largest insurer.

AIG is open to constructive efforts by Mr. Greenberg or others that assist the company in restoring value to shareholders and repaying the taxpayer, AIG said in a statement on Monday.

(Reporting by David Lawder, Lilla Zuill and Rachelle Younglai; editing by John Wallace and Gerald E. McCormick)

Aig