Negotiating a reduction of principal is trying to stick someone else with your debt.
And will thrash your credit for a really long time....as I am finding out (I had "settled" a car loan and a credit card back in 2004). Now I wish I hadn't, (they offered, I didn't ask) as 7 years later it's still hurting my credit, and the difference was only a few thousand, which I would gladly pay now to erase the impact. Morality aside, (again, they offered), it is totally normal for a business to negotiate some principal reduction if it means they will be able to collect 100% of half instead of 0% of the whole thing...it comes down to net present value (money now is worth more than money later), and the ability to write off bad debt as lost revenue.
On the mechanics of this (to the OP), when a settlement is arranged, it is almost always based on a near-term (<30day) lump sum--if they are going to write off part of the debt, they are going to want to maximize both the chance they get the newly negotiated amount and maximize it's NPV. If they are willing to accept the credit rating hit (again, I DO NOT RECOMMEND IT), attempt to get some lower interest rate personal loan or some other method of getting cash BEFORE you offer a settlement, then offer as much as you possibly can as a fixed lump sum.
A better course of action (which won't kill your credit) is before you miss any payments, get a personal loan, balance transfer, sell stuff, whatever (as long as any new money borrowed is at a lower interest rate), then offer a huge payment in exchange for a lower rate on the remaining...this typically works (again, they are trying to maximize their return so big near term $ is worth lowering the rate on the rest, and it reduces their exposure/risk)