Start with the conversation, not the tender
Tell the incumbent what you are doing and why. A supplier who hears about a competitive process from the market has been treated badly. A supplier who is told that the business needs to take cost out, and given a genuine chance to help, often comes back with more than a discount.
Separate price from cost
Unit price is one lever and usually the least interesting one. Specification, volume commitment, delivery frequency, payment terms, packaging, service levels and demand management all change what you pay. Suppliers can often improve total cost without cutting margin, which is a far more durable outcome for both sides.
Do the preparation
Know what the price is made of. Know what the market looks like. Know what you would do if they said no. A conversation where you have done that work is respectful. One where you have not is a request for a favour.
Be honest about the alternative
If you are not going to switch, do not imply that you will. Bluffing works once. What works repeatedly is a clear statement of what you need and what happens if it cannot be found, delivered without theatre.
Give something back
A longer term, faster payment, consolidated volume, forecast visibility, fewer emergency orders. These have real value to a supplier and often cost you very little, which makes them the most efficient currency you have.
Then hold the result
Savings erode through change requests, scope creep and quiet uplifts. Agree how price changes will work, write it down, and have somebody check it twice a year. That protects the relationship too, because nothing sours one faster than a dispute about what was agreed.