I would accept the offer provided that it is a BONA FIDE offer.
Some issues to consider:
1) Make sure the debt is actually yours.
2) Make sure that there is a meaningful period of time left on the statute of limitations (10 years). Because if this "settlement" doesn't work, the clock goes back to time zero for the debt, minus the amount you paid. In other words, don't pay off a debt "by the end of the month" that will expire on the fifteenth.
3) I'm not a lawyer, and therefore, in your shoes, I would consider consulting a lawyer.
4) If using a lawyer isn't cost effective, then I would set up an escrow account, deposit the settlement amount, and then release it upon receipt of a letter that says that the amount in escrow fully satisfies the debt. Such a letter will help your record with credit agencies. It will also establish a record that you've paid, and paid in full. Based on what you've said about the original letter, the debt collector would be willing to do this. If he reneges, he's "playing games."