Judge Robert D. Drain, who is presiding over the cases, has repeatedly refused the banks’ requests to throw out the lawsuits. In July, when he refused to dismiss the case against JPMorgan, he said, “The complaint sets forth a cause of action that Chase is using the inaccuracy of its credit reporting on a systematic basis to further its business of selling debts and its buyer’s collection of such debt.”

At a hearing in April, transcripts show, the judge criticized Citigroup for not changing the way it reports debts to the credit reporting agencies. “I continue to believe there’s one reason, and one reason only, that Citibank refuses to change its policy,” the judge said. The reason, the judge went on, is “because it makes money off of it.”

In a statement, a spokesman for Citigroup said the bank “takes this issue very seriously,” adding that the bank has made a proposal to the plaintiff’s lawyers “consistent” with what the other banks have proposed.

In the hearing this week, lawyers for Citigroup indicated that they were on the brink of making a change similar to what Bank of America and JPMorgan Chase have agreed to, an alteration that could change the credit reports of tens of thousands of people. For many borrowers, the credit report is the difference between getting a job and being turned down.

With so much at stake, borrowers are willing to do almost anything — even pay debts that they worked hard to discharge in bankruptcy.

Diane Torres, who went through bankruptcy in 2010, said she was on the verge of becoming one of the people who paid for debts she no longer owed. The only thing that stopped her, Ms. Torres said, was that she could not afford it.

The problems began, Ms. Torres said, when she applied for a job with a credit union and was told that her credit report showed she had two delinquent accounts — one on a Chase credit card and the other on a credit card from GE Money Bank. Unless she fixed the problem, Ms. Torres said, she would not get the job.