Yes, they do. You can totally resolve your IRS debt for a payment that is a very small percentage of what is owed, if you qualify. The IRS program that allows this is called an Offer in Compromise. Generally, acceptance of an offer in compromise depends on your ability to repay the debt.
If the IRS agrees that the amount offered is the most they will be able to collect on the debt, then they will accept the offer. This generally depends on 4 factors:
1) Income. How much income do you make per month?
2) Expenses. What are your monthly expenses?
3) Assets. What do you own?
4) Liabilities. What debt do you have? What do you owe?
Also, the IRS sometimes will consider extenuating circumstances that affect your ability to pay such as an illness of you, illness of your spouse, etc.