The Treasury could sell a big chunk of its remaining 33 percent stake in General Motors Co in the summer or fall, a person with knowledge of the matter said on Monday, although doing so may lead to more losses for it on the automaker's bailout.

No decision has been made yet on the exact timing or size of such a sale, two sources with knowledge of the matter said.

Sources have previously told Reuters that a follow-on share sale could be worth about $10 billion -- but since GM shares have dropped below their $33 IPO price, some have suggested that the sale could be smaller.

GM's shares are now at their lowest since their IPO. They closed at $29.97, or 9.2 percent below the IPO price on the New York Stock Exchange on Monday.

If the stock doesn't recover, a share sale now would add to the government's losses on the $50 billion it poured into GM during the financial crisis to help the one-time blue-chip avoid liquidation.

The government lost money on the IPO but had hoped to recover the money in follow-on offerings.

The government is willing to take a loss on its investment in GM because the Obama administration wants to end its involvement with the automaker ahead of the 2012 U.S. elections, the Wall Street Journal reported.

The government would need to sell its remaining 500 million or so shares at $53 apiece to break even, the paper reported.

GM is expected to report a profit in the first quarter, the paper said, citing unnamed sources.

June would be the earliest a share sale could take place. It could also happen in the first half of August or after Labor Day, one source told Reuters.

An additional share sale could happen in November or December, that source said.

The information is not public and the people declined to be identified. Treasury declined comment.

Our role is to continue to deliver results and create shareholder value. It is up to the U.S. Treasury as to when they sell their remaining stake, a GM spokesman said.

GM's blockbuster IPO last November reduced the U.S. government's stake in GM to 33 percent of common shares outstanding from the 61 percent it held before the IPO. According to the IPO agreement, the Treasury cannot sell more shares before May 22.

The world's No. 2 automaker has slowly rebounded from its 2009 bankruptcy in part because of new products like the Chevrolet Malibu and Volt which were being developed before its government bailout.

Some investors are concerned about shuffles among top management at GM, where a new chief financial officer took over this month. And GM, top-selling in the U.S. auto market, has had four CEOs since early 2009.

(Additional reporting by Alina Selyukh, Bernie Woodall, Soyoung Kim, Ben Klayman and Paritosh Bansal; Editing by Lincoln Feast)