The short answer: 660+ for a decent rate, 700+ for a good one, and 740+ for the best terms available.
Most online lenders have a minimum credit score requirement between 580 and 660. But qualifying and getting a rate that actually saves you money are two different things. If you're approved at 580 with a 25% interest rate, you haven't improved your situation at all if your credit cards are at 24%.
Here's roughly what to expect by score range:
740+: Rates of 7% to 12%. These are the loans worth getting excited about. Major savings on credit card debt.
700-739: Rates of 12% to 17%. Still a meaningful improvement over most credit cards.
660-699: Rates of 16% to 22%. Marginal benefit. Run the numbers carefully before committing.
Below 660: Rates of 22% to 36%. At these rates, a consolidation loan often doesn't save enough to justify the origination fees and hard credit pull. Consider a debt management plan or credit counseling instead.
Credit unions are often more flexible than online lenders, especially if you have an existing relationship. Some credit unions will work with scores in the low 600s and still offer rates in the 12% to 16% range.
Before applying anywhere, check your free credit reports at AnnualCreditReport.com and dispute any errors. A corrected error could bump your score by 20 to 50 points, which might move you into a better rate tier. Also, check if any lenders offer pre-qualification with a soft pull so you can see estimated rates without impacting your score.