With a deadline looming on Monday for state officials to sign onto a landmark multibillion-dollar settlement to address foreclosure abuses, the Obama administration is close to winning support from a crucial state that would significantly expand the breadth of the deal.

The biggest remaining holdout, California, has returned to the negotiating table after a four-month absence, a change of heart that could increase the pot for mortgage relief nationwide to $25 billion from $19 billion.

Another important potential backer, Attorney General Eric T. Schneiderman of New York, has also signaled that he sees progress on provisions that prevented him from supporting it in the past.

The potential support from California and New York comes in exchange for tightening provisions of the settlement to preserve the right to investigate past misdeeds by banks, and stepping up oversight to ensure that the financial institutions live up to the deal and distribute the money to the hardest-hit homeowners.

The settlement would require banks to provide billions of dollars in aid to homeowners who have lost their homes to foreclosure or who are still at risk, after years of failed attempts by the White House and other government officials to alter the behavior of the biggest banks.

The banks  led by the five biggest mortgage servicers, Bank of America, JPMorgan Chase, Wells Fargo, Citigroup and Ally Financial  want to settle an investigation into abuses set off in 2010 by evidence that they foreclosed on borrowers with only a cursory examination of the relevant documents, a practice known as robo-signing. Four million families have lost their homes to foreclosure since the beginning of 2007.

As recently as two weeks ago, with federal officials hoping to complete a deal that President Obama could cite in his State of the Union address, Californias attorney general, Kamala Harris, made it clear she was not on board, terming the plan inadequate. But in the last few days, differences have narrowed in negotiations that one participant described as round the clock, with California officials in direct communication with bank representatives for the first time in months.

For the past 13 months we have been working for a resolution that brings real relief to the hardest-hit homeowners, is transparent about who benefits, and will ensure accountability, Ms. Harris said in a statement. We are closer now than weve been before but were not there yet.

The settlement has been hamstrung by one delay after another over the last year. Winning Californias support now would represent a major win for the White House in this election year.

I am encouraged by the conversations weve had with many states in the last few days, said Shaun Donovan, the secretary of housing and urban development. This will be one of the most significant steps in the recovery of homeowners, neighborhoods and the broader housing market from the worst collapse since the Depression.

My fundamental point is that its a first step, he added, citing measures like Mr. Obamas proposal last week to lower interest rates for homeowners who are still current on their mortgages.

Officials involved in the negotiations cautioned that broader state support could still be days away. And although the timing of any announcement is subject to last-minute maneuvering, as it stands now the deal would set aside up to $17 billion specifically to pay for principal reductions and other relief for up to one million borrowers who are behind on their payments but owe more than their houses are currently worth. The deal would also provide checks for about $2,000 to roughly 750,000 who lost homes to foreclosure.

Those figures are contingent upon the number who respond to the offer, which is likely to go to people who lost their homes between Jan. 1, 2008, and Dec. 31, 2011. In addition, said Patrick Madigan, the Iowa assistant attorney general, homeowners who participate in the settlement will still have the right to sue the banks for improper behavior in the foreclosure process.

California has been focused on measures that would benefit individual homeowners, while New York has been most interested in preserving its ability to investigate the root causes of the financial collapse.

Another critical issue for California is narrowing the amnesty given to banks because under the states False Claims Act, state officials and huge pension funds like Calpers would be able to collect sizable monetary damages from the banks if they could prove mortgages were improperly packaged into securities that later soured. What is more, Californias participation would result in having more money available for many other states, including an estimated $500 million in additional money for Florida.

But the agreements terms do not guarantee minimum allocations of mortgage relief by state.

Mr. Donovan added that there had been numerous discussions with individual states that had specific concerns.

California officials and other veterans of the foreclosure crisis are haunted by the failure of past attempts to alter the behavior of the big banks, including a 2008 deal with Countrywide Financial, the subprime giant now owned by Bank of America, and a more recent agreement last April between federal regulators and the biggest mortgage servicers.

The backers of the latest deal insist their plan has more teeth, with a powerful outside monitor to oversee enforcement and heavy monetary penalties if banks fail to live up to commitments. While the past agreement with Countrywide gave banks credit even if their offers to modify the interest rate of the mortgage or write down principal were not accepted by borrowers, this deal counts only what banks actually do for homeowners.

If banks fall short of the multibillion-dollar benchmarks set out for principal reduction and other benefits for homeowners, they will have to pay the difference plus a penalty of up to 40 percent directly to the federal government, according to Mr. Madigan.

The depressed housing market continues to pose a drag on the halting economic recovery. RealtyTrac, which analyzes housing data, predicts two million more foreclosures over the next two years. Some 11 million families owe more on their houses than they are worth.

The settlement, if all states participate, will also include $3 billion to lower the rates of mortgage holders who are current. Banks will get more credit for reducing principal owed and helping families keep their homes, and less for short sales or taking losses on loans that were likely to go bad, like those that were severely delinquent.