A debt may be collected by the original creditor, it may be sold to a factor (in simple terms, a company that buys debts that are believed to be collectable), or it may sell old, dubious debt to a junk debt buyer. Once a debt is sold, the money is owed to the buyer of the debt.

If a debt buyer has a policy to always take money from a debtor, no matter how little, then its agents would be expected to follow that policy at risk of job discipline. In some states, accepting a payment can potentially provide a basis to overcome a statute of limitations defense, and there's really no reason not to accept a voluntary, unconditional payment from a debtor. But no creditor has to enter into a payment plan on a debt that is due in full and a debtor has no recourse against a collection agent who refuses to negotiate a payment plan on such a debt. Creditors don't have to offer written confirmation of repayment plans, but I would be skeptical as a debtor of entering into a repayment plan (particularly if it's to settle a lawsuit or for forebearance of a lawsuit) if I did not have the agreement in writing.