Yes — in many situations lenders will negotiate. This can happen through a short sale, where the lender accepts less than the full mortgage balance to release the property, or through a deed in lieu of foreclosure, where the homeowner signs the property over to the bank in exchange for forgiveness of the remaining debt. Lenders often prefer these options over a lengthy foreclosure, which is expensive and slow for them too.
To negotiate successfully, the homeowner usually needs to demonstrate genuine financial hardship and show that the current property value does not cover what is owed. Having documentation organized — income statements, hardship letter, recent property valuation — and understanding the process up front meaningfully improves the chances of a favorable outcome. This is one of those situations where knowing your options early gives you significantly more leverage than waiting until things become urgent.