I agree with my colleagues. There are mortgage companies that will agree to a forbearance agreement for a short time while you are unemployed. However, keep in mind that mortgage companies have no obligation to modify your mortgage, or offer a trial period. Mortgage companies can keep you to your original agreement and file for foreclosure if that agreement is broken.
Without any income you have zero ability to pay. Unfortunately, while a temporary delay in foreclosure may be possible, without regular income a modification is a dead end. Focus on recovering your income as quickly as possible.
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There are some mortgage modification programs available for people on unemployment. Typically, the individuals that answer the telephone at the mortgage servicer won't tell you about these programs because they either don't know (poorly trained) or don't have the discretion to provide that information. You need to escalate your denial to a manager, tell them you are receiving unemployment and that you want to know what programs are available for someone receiving unemployment because you want to apply.
If you stil hit a brick wall, then I would strongly urge you to consult with a housing counselor trained in mortgage modifications or an attorney, but bear in mind that there are alot of scam artists out there right now, so be careful. You should run away from anyone who guarantees you a mortgage modification for a large "up front" retainer fee.
Finally, make sure you keep good WRITTEN records of all contacts made with the mortgage servicer. Under New York State law, mortgage servicers are required to pursue a mortgage modification for you in "good faith." If you ever get into Court, you may need these records to demonstrate to a Judge that they failed to meet this requirement.