You say your primary goal is to clean up your credit report, and you're willing to spend some cash to do it. OK. But beware: the law in this area is a funhouse mirror, everything works upside down and backwards.
To start, let's be clear: Credit reports are not extortion to force you into paying. They are a historical record of your creditworthiness, and almost impossible to fix without altering history.
Paying on this debt will affirm the old data was correct, and glue it to your report.
Do NOT pay the debt. Do NOT acknowledge the debt.
Here's how credit reporting works for R-9 (sent to collections) amounts. The data is on your credit report for 7 years. The danger is in this clock being restarted.
What will not restart the clock? Ignoring the debt, talking casusally to collectors, and the debt being sold from one collector to another.
What will restart the clock? Acknowledging the debt formally, court judgment, paying the debt, or paying on the debt (obviously, paying acknowledges the debt.)
Crazy! You could have a debt that's over 7 years old, pay it because you're a decent person, and BOOM! Clock restarts and 7 more years of bad luck. Even worse-- if they write-off or forgive any part of the debt, that's income and you'll need to pay income tax on it. Ugh!
Disavow, but settle
Like I say, the only way to remove a bad mark is to alter history.
Simple fact: The collector doesn't care about your bad credit mark; he wants money. And it costs a lot of money, time and/or stress for both of you to demand they research it, negotiate, play phone-tag, and ultimately go to court. So this works very well (this is just the guts, you have to add all the who, what, where, signature block, formalities etc.):
1 Company and Customer absolutely disagree as to whether Customer owes
Company this debt: (explicitly named debt with numbers and amount)
2 But Company and Customer both eagerly agree that the expense, time, and
stress of research, negotiation, and litigation is burdensome for both of us.
We both strongly desire a quick, final and no-fault solution. Therefore:
3 Parties agree Customer shall pay Company (acceptable fraction here).
Payment within 30 days. To be acknowledged in writing by Company.
4 This shall be absolute and final resolution.
5 NO-FAULT. Parties agree this settlement resolves the matter in good faith.
Parties agree this settlement is done for practical reasons, this bill has not been established as a valid debt, and any difference between billed and settled amount is not a canceled nor forgiven debt.
6 Neither party nor its assigns will make any adverse statements to third parties relating to this bill or agreement. Parties agree they have a continuous duty to remove adverse statements, and agree to do so within seven days of request.
7 Parties specifically agree no adverse mark nor any mark of any kind shall be placed on Customer's credit report; and in the event such a mark appears, Parties will disavow it continuously. Parties agree that a good credit report has a monetary value and specific impacts on a customer's life.
8 Jurisdiction of law shall be where the effects are felt, and that shall be (place of service) regarding the amounts of the bill proper.
Severable, inseparable, counterparts, witness, signature lines blah blah.
A collector is gonna sign this because it's free money and it's not tricky.
What does this do? 1, 2 and 5 alter history to make the debt never have existed in the first place. To do this, it must formally answer the question of why the heck would you pay a debt that isn't real and you don't owe: out of sheer practicality; it's cheaper than Rogaine. This is your "get out of jail free" card both with the credit bureaus and the IRS.
Of course, 3 gives the creditor motivation to go along with it.
6 says they can't burn your credit. 7 says it again and they're agreeing you can sue for cash money. 8 lets you pick the court. The collector won't get hung up on any of these since he can easily remove the bad mark. (don't be mad that they won't do it "for free", that's what 3 is for.)
The key to getting them to take a settlement is to be reasonable and fair. Make sure the agreement works for them too. 6 says you can't badmouth them on social media. 4 and 5 says it can't be used against them. 8 throws them a bone by letting them sue in their home court for the bill they just settled (a right they already had). If it's medical, add "HIPAA does not apply to this document" to save them a ton of paperwork. Make it easy for them.
You want the collector to take it to his boss and say "this is pretty good. Do it." Don't send the money until their signed copy is in your hands. Then send promptly with an SASE for the receipt. Make it easy for them.
This is on you. As far as "getting them to send you an offer", creditors are reluctant to mail things especially to people they don't think will pay, because it costs them money to write and send. So you may need to be proactive about running them down with your offer. Like I say, it's a funhouse mirror.