Honestly, you are probably not going to get anywhere "negotiating" a payoff amount less than what you owe with federally guaranteed student loans (the vast majority are of this variety). A totally private loan? Maybe. But not a federal loan. You could certainly consolidate and lower your rate, but not negotiate a lower payoff.
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It is almost impossible to negotiate the student loans away. Good Luck.
Disclaimer: This answer does not constitute legal advice. I am admitted in the States of New York, New Jersey and Massachusetts only and make no attempt to opine on matters of law that are not relevant to those three States. This advice is based on general principles of law that may or may not relate to your specific situation. Facts and laws change and these possible changes will affect the advice provided here. Consult an attorney in your locale before you act on any of this advice. You should not rely on this advice alone and nothing in these communications creates an attorney client relationship.
If you have not already consolidated your loans, try calling your lender and asking about consolidation. Sometimes by consolidating you can get a better rate, which could in turn lower your monthly payment.