But the letters sent by Chase and Bank of America clearly warn that the forgiveness will be reported to the I.R.S. If so, these borrowers may have to prove that the banks erred in claiming to have forgiven the debts.

I ASKED spokesmen for Chase and Bank of America how they could forgive debts that no longer existed. Both gave the same unsatisfying answer. Very similar letters had been sent, both banks said, to two very different types of borrowers. One set of borrowers has outstanding debt that the banks are offering to forgive. The other set has had their debts discharged in bankruptcy, but the bank still holds a lien against their properties. Releasing the liens provides a benefit to borrowers when they go to sell their homes, and both banks said the letters were intended to notify borrowers whose liens were being released.

Why not take care to write letters specifically tailored to each borrower’s situation?

Dan Frahm, a Bank of America spokesman, said the bank would work on clarifying what was in the letters to borrowers. And, late Friday, the bank put a more extensive description of the forgiveness and lien release program on its Web site. Not a bad idea, since nowhere does Bank of America’s letter discuss releasing the lien. Mr. Frahm estimated that 12,000 Bank of America customers whose debts had been discharged had received these letters.

Tom Kelly, a Chase spokesman, conceded that the bank “may have caused some confusion for customers.” Its letter does note that the bank is releasing the lien on the property.

But even this is incorrect in Ms. Esposito’s case, Mr. Crane said. Her lien was actually eliminated back in 2009, during her bankruptcy proceeding.

All of this made me wonder: are the banks’ forgiveness letters a way to gain credits for debts these institutions are improperly claiming to have extinguished? The banks say no.

But Chase appears to be claiming to release a lien on Ms. Esposito’s property that it does not hold. And under the mortgage settlement, it could receive a credit.