Thank you.If the hospital sold the debt to a collector then the hospital no longer owns it. You should be paying whoever the collector is that bought the debt. I'd try to get something in writing for whatever payment terms you can come to agreement on. Make sure it includes them reporting to the credit bureau that the debt is "paid in full". If they want to list it as something else (ie paid as agreed) make sure you understand what it means for your score. Also, remember in your negotiations that they paid pennies on the dollar to take over your debt and if it has been resold more than once the amount paid continues to decrease. You should be able to pay far less than the listed amount.
Of course all this is predicated on you not being outside the statute of limitations. If the statute of limitations has expired don't even acknowledge the debt if they call you. As soon as you do they will try to claim it restarts the clock.
I have five hospital bills in collections. Three of them with one collection bureau. When I tried to work with them in 2013, they wanted $65 per month per bill. I told them I could do $25 per month (I was already in the hole every month as it was). They refused. So, we will see. :/