Two years after General Motors Co. and the Obama administration worked together to save the auto maker, the two are now at odds over how to get the government out of the company.

The Detroit auto maker wants a share sale as soon as possible, people familiar with the matter said, while the administration has a goal of selling its remaining 27 percent stake in August or September, after GM's next quarterly results are released, reported The Wall Street Journal.

The impasse is hardening seven months after the two sides celebrated a public stock offering that returned GM to Wall Street and brought in $13.5 billion for U.S. taxpayers.

Both are eager to cut those ties. GM says pay limits imposed by the government hamper its ability to recruit top talent. The Obama administration wants its still-debated ownership to be over by the time campaign season starts to heat up, people familiar with the matter said.

But the U.S. Treasury's exit plan has become less clear as GM shares remain below the $33 price of GM's November IPO. Shares closed up 38 cents at $29.97 in 4 p.m. New York Stock Exchange trading on Wednesday. However, the price is lower than either Wall Street or Treasury officials had anticipated. Late last year, some Wall Street analysts predicted GM's share price could rise to between $40 and $50 within the year.

The issue flared up Wednesday with Republican lawmakers blasting the bailout in a hearing on the effects of GM's bankruptcy.

The auto-industry rescue has been central to President Barack Obama's message as he heads into next year's re-election campaign. If Treasury takes a larger-than-expected loss on GM or remains stuck with its GM stake, it could hurt the narrative of Mr. Obama's campaign.

The U.S. so far has recouped $27 billion of the $50 billion it spent on the bailout; any additional recovery would be through a share offering.

Treasury could delay an offering, or opt for a relatively small share sale if the share price remains depressed. Earlier this year, the company approached Treasury officials about buying back shares directly from the U.S. At the time, Treasury officials said they wouldn't consider that option, these people said. Now, GM is considering a plan to buy back shares from existing stockholders, people familiar with the matter said. A buyback could increase the value of outstanding shares and indirectly aid the Treasury.

At $30 a share, taxpayers would lose about $12 billion on the GM rescue, a bigger loss than the $7 billion they would face if shares rose to the $40 level that some analysts had expected. A recent White House report put the total loss to taxpayers at $14 billion out of $80 billion bailout of the industry, down from its initial estimate of $48 billion.

On Wednesday, one of the architects of the bailout, White House adviser Ron Bloom, testified before a mostly-Republican Congressional panel.

"The administration absolutely picked winners and losers in the bailout—the losers were the American taxpayers," said Rep. Mike Kelly, a Pennsylvania Republican whose Cadillac dealership was almost shut down in the GM restructuring.

Mr. Bloom defended the rescue. "The entire ability of the United States to make cars was at risk as this point," he said. "What the American taxpayers get is the fact that they have an automobile industry."