IRC s. 61(a)(12) defines discharge of indebtedness as income. Loan proceeds are not on s. 61's list, so they're not income. But when you're not going to pay them back, then they're income.
That being said, the penalty on the lender for not doing a form 1099-c is $50 if they're caught. So raising your offer by $50 should be more than enough to get them to decline to issue one. A recitation that the debt was doubtful and disputed inserted into the settlement agreement should be enough to immunize both sides from any further inquiry.
Remember, it's a hassle for them to prepare it and issue it ... they're hard to figure correctly and the collectors that actually issue them are usually large Junk Debt Buyers that the IRS watches closely, or collectors that you have annoyed mightily so that they do it on a retaliatory basis.
A form 982 can reduce or eliminate the effect of a 1099-c form, provided you're really more or less broke.
Mr. Hankins makes an important point about the settlement agreement. It needs to include language that indicates you gave up something of value in exchange for the lower settlement amount so that none of the debt was forgiven. You may have given up valuable defenses, counterclaims, FDCPA violation claims, etc. You don't have to specify and you don't have to put a dollar amount on them. The key is for the creditor and debtor to agree there was an exchange of value that satisfied the debt.
Good advice above, let me put my 2 cents in ( whichisn't income). The trigger is if the amount of principal forgiven is $600+, so you may or may not have to worry. If it is a bank, you will probably get a 1099 if it is a large discount. If it is a Junk Debt Buyer, you can probably negotiate your way out. If you have been sued, have defended the suit and then you settle, no 1099 should issue. If it is a collcetion agency, well, I am not in favor of paying collection agencies as they cannot bind their clients to anything.