The U.S. Treasury Department on Thursday announced plans to sell 410,000 shares in auto lender Ally Financial as part of its effort to unwind its financial bailout fund.

The Treasury said it expected taxpayers to recover about $3 billion from the private offering of Ally common stock at $7,375 per share. The sale would reduce the government's stake to 37 percent, it said.

The government pumped $17.2 billion into Ally during the 2007-2009 financial crisis, and the Treasury said taxpayers will have recovered about $15.3 billion once the stock sale was complete.

The offering could also leave the Treasury just a few billion short on the investments it made to prop up lenders, automakers and the housing sector from its crisis-era $700 billion Troubled Asset Relief Program.

Private market investors appear to be optimistic about Ally's prospects. Ally sold $1.3 billion in unlisted shares to private investors in November for an average price of around $6,000 per share, and weeks later GM sold its remaining 8.5 percent stake in Ally for around $6,800 per share.

The appetite for Ally shares from private investors suggests the government might be able to fully exit its stake this year.

In announcing its offering, the Treasury said it would work with Ally to further reduce the government's investment through either a public offering, private sale of its common shares or other alternatives.

Ally has been hoping to go public since at least 2011, but investors had remained worried about the problems that forced it to seek a taxpayer bailout, which included bad home loans at its Residential Capital subprime mortgage unit.

That unit is now getting ready to emerge from bankruptcy.