If I try to get a loan from a bank or credit union, will they negotiate on the interest rate?

Remember that the INTEREST RATE may be much less important than the TERMS.
Credit card lenders state that they can change the interest rate and fees at any time for any reason.
You don't want that.
You want a fixed rate with little or no penalty for minor late payments except paying the extra interest.
It also depends on what you want the loan for and how much you want.
If you want $40,000 to improve a house that you bought in 1980 for $70,000, own free and clear, that is currently appraised for $500,000, and you are employed by the State for the last 15 years at $100,000 per year with no other debt, you should get an excellent fixed rate with good terms.
If you want $4,000 to go on vacation and you earn $20,000 per year doing freelance landscaping and you have $15,000 credit card debt you will get a terrible rate and terms if anyone lends to you at all because you shouldn't be borrowing money in the first place.
But, yes, they will negotiate. Just keep an eye on the terms.

In most cases, the answer is yes. What rate you get is heavily dependent on your credit rating: A judgment of someone's ability to repay debts, based on current and projected income and history of payment of past debts. Sometimes expressed as a number called a credit score (see glossary in Finance>Loans>Credit Cards for other helpful terms). It is always wise to shop around for the best interest rate.

For example, if you have no credit, have only had credit for a short period of time, or a lot of negative avtivity (late payments, bankruptcy, etc.) you will likely have less room or no room to negotiate the rates.

If, however, you have had credit for a long time and are current on all your payments then you will have room to negotiate the rates.