A settlement is a private agreement between two parties with a dispute. There is no legal basis to force someone to agree to a settlement. A creditor is well within their rights to demand full payment, they don't have to settle.
They do not have to accept your offer. Of course, if you are paying on the debt, the likelihood of them suing you decreases.
This post is for information purposes only and does not constitute legal advice, nor does it establish an attorney client relationship with Mr. Cassara.
I would agree with the two attorneys who have previously answered this question, and would add that you should request they provide you with a written statement regarding any extra fees they want to tack on.
My answer is for general discussion purposes only. It is not intended to be legal advice regarding your specific situation and does not create an attorney-client relationship.