Legally a debt collector CAN NOT speak to anyone other than the debtor, their spouse or their attorney regarding the details of a debt they are attempting to collect. They can however call your friends, co-workers and other family members to gain current telephone and address information only, but that is it. They can only call them once and they MUST not discuss any details regarding your debt. By them asking to speak to your supervisor is just their way of intimidating you into believing they are going to disclose the debt and embarras you in front of your boss and co-workers. However, they can not do that and if they do, it is illegal and you should report them to the BBB, your attorney genernal and even maybe seek advice from a private attorney.
To answer your other questions, since they offered you a settlement, ask them to send it in writing in addition to the debt validation information. Once you receive it and verify the debt is valid and they have the right to collect it, then contact them and work out payment arrangements that you can afford. They are going to tell you that they will need all the money at once and if you cant do that, then the settlement offer will be voided and you will be responsible for the entire amount. They will probably also tell you that they are going to garnish your wages, if they tell you this, then they are implying that they are going to take you to court and get a court order for garnishment. They can only tell you this if they actually intend to do so and have legal right to do so. The question is if they (the debt collector) have legal right or not. I always thought the original creditor only had the right to sue. If this is the case, then Discover could take you to court and you will be required to appear. You would want to appear in this case because you may have a chance at getting the debt reduced. Besides, you might get in trouble if you are served and dont answer or appear when ordered.
If you want them to stop bothering you at work, you can stop them by sending a certified letter on your company letterhead stating you are not allowed to receive personal calls at work and to immediately cease all calls to your place of employment. By law, they must cease all calls after receipt of the letter. If they dont, then again, report them to the BBB, your attorney general and even a private attorney.
Good luck and I hope this helps.