You are lacking a lot of information needed to answer your question. The best advice would be to consult with a local attorney. Bring all documentation and information with you for them to review. Let them negotiate any type of settlement on your behalf. Most attorneys offer a free consultation prior to accepting your case.
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Debts do not disappear after the SOL and reporting limits elapse. Amex can continue collections and can resell the debt. They can also deny extending credit to you in the future. If they cannot give you the deal that you want right now, why not try again sometime in the future. Is there any real impetus to settle right now?
This is not legal advice. Seek a consultation with an attorney in your jurisdiction. An attorney/client relationship is not sought or formed by the participation in this website. I am licensed to practice law in New York and New Jersey.
Just remember that they can always hound you, but nobody asking for Amex can threaten to sue you for it. That is, collection agencies or lawyers. Just monitor your credit reports. Even wonderful people like Amex have been known to futzing with the last activity date just to keep it in your report. If you see such actions, see a good consumer attorney