Very generally speaking, the solution you describe is possible. With that said, the debt collector will probably continue to badger you until the debt is paid in full, even if you make promises to pay in the future. They will also probably want some assurance more than your word that the payment will be made.
If you call the debt collector, make sure you tell them you wish no further contact by mail or phone. Set up a payment plan as you propose and stick to it. Get the name, phone number and address of the person with whom you spoke and try to deal only with that person. Otherwise, you will continue to be bothered by the agency.
If possible, get this agreement in writing. They may very well not want to do this. The bottom line is, the only way to answer your question is to ask the debt collector. The worst they can do is say no. And then it will probably take them more than the 90 days to get a lawsuit filed, court date, etc. As previously advised, keep track of every letter and telephone call--names, dates, times, what was discussed. A federal law called the Fair Debt Collection Practices Act may give you some leverage if they try to play games.
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(1) Call them back and tell then you are not disagreeing with the amount owed.
(2) Explain your side truthfully and tell them what you can realistically afford and that you want a payment plan.
(3) I've never turned down anyone who wants to resolve something within 90 days and I've dealt with thousands of people in your situation. If they don't want to resolve it within 90 days they are being very unreasonable.
You're in good shape, they don't WANT to take you to court if you can resolve it.