But now banks, eager to get Washington out of their hair, are pushing to undo those investments as quickly — and cheaply — as possible. If the Obama administration acquiesces, billions of taxpayer dollars could be left on the table, The New York Times’ Eric Dash writes.

At issue are so-called warrants that the government received from the banks last autumn, when the financial world was teetering. Like options, warrants give their owners the right to buy stock at a set price over a certain period of time, in this case, 10 years.

Now, with many banks itching to return their bailout money, the warrants are raising some thorny questions. What are these investments worth? Should the government drive a hard bargain, or let the banks off easy? Should it maximize profit for taxpayers, or minimize pain for banks?

Many banks want to buy back the warrants and wriggle free of the government. Big banks like JPMorgan Chase, Goldman Sachs and Morgan Stanley have formally notified regulators that they want to return their bailout money, The Times said, citing people briefed on the situation. But as long as the government holds the warrants, it still has some leverage over the industry.

For taxpayers, a lot of money is at stake. The government has an option to buy stock in 579 banks. By some estimates, the warrants on JPMorgan alone are currently worth more than $1.1 billion. They could be worth much more if JPMorgan’s share price rises.

So far, one publicly traded bank, Old National Bancorp in Indiana, has repaid the government in full by returning its bailout money and repurchasing its warrants. (Two small privately held banks have done the same.)

How Old National pulled this off, and the seemingly favorable terms it secured, shows how aggressively banks big and small are pushing, even after they repay money from the Troubled Asset Relief Program, or TARP. Old National paid $1.2 million for its warrants. Analysts estimate the investments might have been worth as much as $6.9 million.

“It’s a great deal for Old National,” Linus Wilson, a finance professor at the University of Louisiana, Lafayette, told The Times. “Treasury accepted a lowball offer.”

Andrew Williams, a Treasury spokesman, declined to comment about the negotiations, but told The Times that the government “has a robust process in place for valuing warrants.” He added that the Treasury was required by law to sell the warrants once a bank repaid its bailout money.

Analysts say that has made it difficult for the government to pursue a goal of maximizing profits for taxpayers, though a recent change to the law might give the Treasury more flexibility. Even if it had the option, it is unclear how successful the government would be at actively managing such a complicated investment portfolio.

Mr. Williams said the Treasury’s total warrant holdings were worth more than $5 billion, but the value changes along with the underlying stock prices and other factors.

But Prof. Wilson estimated that the warrants on nearly 300 publicly traded banks, which account for more than 95 percent of the government’s investments, were conservatively worth $2.4 billion to $10.9 billion. Some lawmakers worry that taxpayers will lose out. “Taxpayers were there at a critical moment,” Senator Jack Reed, Democrat of Rhode Island and a member of the banking committee, told The Times. “They should enjoy the upside when these institutions recover.”

To extinguish the warrants, the banks can let the Treasury auction them off to private investors or can choose to buy them back themselves. As with other bank rescue efforts, like moves to buy banks’ problem assets, the central issue with the repurchases is determining a fair price.

“That is the problem in TARP asset purchases, and it is the problem here,” Vincent R. Reinhart, a former Federal Reserve official who is now a resident scholar at the American Enterprise Institute, told The Times. “Do you value it at roughly comparable asset prices or do you acknowledge that the current market prices reflect an unusual uncertainty in markets and aversion to risk?”

For the government, the decision is about more than dollars and cents. It may be willing to sell the warrants simply to send a positive message about the stability of the banks.

“The U.S. Treasury would be better off rejecting a lot of these bids and selling these warrants to third-party investors,” Prof. Wilson, at the University of Louisiana, told The Times. “Instead of having one buyer, they would have many buyers from all over the world trying to decide what the proper price should be.”

Old National’s move to buy back its warrants illustrates how tricky it is to strike the right balance. Executives at the bank, based in Evansville, Ind., and a large community lender with 100 branches and $8 billion in assets, began seeking to exit TARP almost as soon as the Treasury wired it the money in December.

By the end of March, Old National had won approval from its regulators to repay its $100 million of bailout funds. But the bank also wanted to repurchase its warrants, fearing it could remain subject to pressure from the government or another outside investor.

“We felt more comfortable that we controlled our own destiny rather than have the hands of the Treasury or a third party,” Bob Jones, Old National’s chief executive, told The Times. “We think our stock has plenty of upside and would rather have that in our hands.” On April 20, Old National submitted an initial offer of around $600,000. Ten days later, Treasury officials, after gathering their own estimates from two asset managers and two market participants, rejected the bid as too low. Over the next week, both sides haggled over the price.

“We both walked through where we were,” Mr. Jones said. “They held their ground on a number of cases, and clearly we had to compromise.” On May 7, Old National was given approval to buy back its warrants for $1.2 million. The bank wired the money four days later.

At Old National’s annual meeting, shareholders were elated by the news. “It was the only applause I drew the whole meeting,” Mr. Jones said.