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Bank of America and federal prosecutors have accelerated their negotiations to resolve an investigation into the bank’s sale of troubled mortgage securities before the financial crisis. The two sides, however, remain far apart on crucial issues and a settlement remained elusive late Wednesday, even after the bank significantly raised its offer.

The bank’s lawyers and Justice Department prosecutors met in Washington on Wednesday to discuss the size of a potential cash penalty, a major sticking point in the settlement talks, according to people briefed on the meeting. Heading into the meeting, the Justice Department was demanding roughly $17 billion to settle the case, more than $10 billion in the form of a cash penalty and the rest in so-called soft dollar payments to help struggling homeowners.

The bank was offering a total of $13 billion, the people said, including $4 billion in cash. The bank narrowed the gap on Wednesday, the people said, raising its cash offer to about $7 billion and its total proposal to roughly $14 billion.

But the Justice Department, which has measured the success of its mortgage cases largely on the size of cash penalties, has balked at the offer. If a deal is not reached in the coming days, the Justice Department might announce a lawsuit against the bank.

Underscoring how little leverage the bank has in fighting the Justice Department, Judge Jed S. Rakoff of the Federal District Court in Manhattan ordered Bank of America on Wednesday to pay a nearly $1.3 billion penalty in another federal mortgage case. The ruling comes nine months after federal prosecutors persuaded a jury to find Bank of America liable for selling questionable loans to Fannie Mae and Freddie Mac, the government-controlled mortgage finance giants, before the financial crisis. Bank of America refused to settle the case and went to trial, a roll of the dice that came back to haunt the bank and could now bleed into negotiations in Washington.

That case, and the separate mortgage settlement talks underway in Washington, further tarnish Bank of America as a symbol of all that was bad in the mortgage market leading up to the foreclosure crisis. Many of the problems stem from Countrywide Financial, the large subprime lender that Bank of America acquired in early 2008.

The case in Manhattan exposed fraudulent practices in one of Countrywide’s lending programs nicknamed the hustle. Federal prosecutors in Manhattan had argued that the hustle program, which linked bonuses to how fast bankers could originate loans, led Countrywide to “cut corners” as it installed “unqualified and inexperienced” loan processors and tore down internal controls that were in place to root out risky borrowers.

“It was from start to finish the vehicle for a brazen fraud by the defendants,” Judge Rakoff wrote in a 19-page opinion, “driven by a hunger for profits and oblivious to the harms thereby visited, not just on the immediate victims but also on the financial system as a whole.”

The ruling from Judge Rakoff, an outspoken critic of financial fraud and the government’s uneven efforts to punish it, came as Bank of America and the Justice Department tried to hash out the broad parameters of a possible deal on Wednesday.

The two sides also gathered for a lengthy negotiating session on Tuesday that focused on the homeowner relief portion of a potential deal and a statement of facts that would outline the bank’s misconduct related to the sale of mortgage securities, the people briefed on the matter said.

The discussions, the people said, focused partly on how to distribute the soft dollar payments. The payments would flow to some familiar causes.

Bank of America, like Citigroup and other banks that settled mortgage security cases, would lower the balances of existing mortgages and help restore vacant properties. But the money would also flow to some more novel ones.

The Justice Department, for example, suggested steering some relief to pension funds and other public investors that suffered losses on the mortgage securities, while the bank has floated a plan to buy back more than 150,000 troubled mortgages from investors.

Buying the mortgages would theoretically give Bank of America more control over how it could modify the loans to help struggling homeowners. The move would allow the bank to get around federal rules that often restrict principal reductions on mortgages backed by Fannie Mae, Freddie Mac and the Federal Housing Administration.

On the surface, the proposal might seem like a costly concession for the bank and a victory for the Justice Department. The bank would plan to resell many of the loans after it modifies them, a person briefed on the matter said, but for a time it would be saddled with an influx of delinquent loans, potentially totaling billions of dollars.

The proposal also seems meant to mollify critics of the Justice Department, who have charged that previous mortgage settlements with banks have fallen short of providing meaningful relief to homeowners.

Since the depths of the foreclosure crisis, consumer advocates, Wall Street watchdog groups and some members of Congress have pushed the banks to forgive more mortgage principal — a move that can bring immediate relief to underwater borrowers.

Yet as of Wednesday, the buyback idea seemed like a long shot. The Justice Department has pointed out a number of practical hurdles to the mortgage buyback plan, the people said, including whether the bank can actually track down all of the homeowners and whether the government can make sure the bank pays a fair price in acquiring the loans.

In some cases, the Justice Department noted in discussions with the bank, investors might not be willing to part ways with the loans, some of which may have been gaining in value as the housing market improves. And the bank could end up profiting if it buys the loans at a steep enough discount and then resells them later.

This week’s discussions follow weeks when talks had essentially been frozen, while the Justice Department worked out a $7 billion settlement with Citigroup.

In that case, the Justice Department went so far as informing Citigroup’s lawyers that it was filing a lawsuit after the bank refused to raise its cash offer. Citigroup’s lawyers had argued that the cash penalty should be based on the bank’s relatively small share of the mortgage securities market in the years before the financial crisis — an argument the Justice Department flatly rejected.

Tensions over the cash penalty have also shaped Bank of America’s negotiations. The bank has balked at paying a large penalty for the defective mortgage securities sold by Countrywide and Merrill Lynch before Bank of America agreed to acquire the companies in 2008, according to people briefed on the matter.

But Judge Rakoff’s ruling, which imposes a nearly $1.3 billion penalty on Bank of America for fraud committed by a Countrywide mortgage program, undercuts that argument.

“Today, Judge Rakoff imposed stiff penalties in a case brought by this office to punish and deter the fraudulent and reckless lending activities of a financial institution leading up to the financial crisis in 2008,” Preet Bharara, the United States attorney in Manhattan, said in a statement on Wednesday.

In a statement about the ruling, Bank of America said: “We believe that this figure simply bears no relation to a limited Countrywide program that lasted several months and ended before Bank of America’s acquisition of the company. We’re reviewing the ruling and will assess our appellate options.”