This is the best tool: A straight reamer with a pilot. It makes a clean counterbore of the exact size, with no splitting or chipping. I've made them up in many different sizes for different Luthier jobs. I have a couple sized just for bass tuner bushings. I normally use them in a drill press, slow speed, but they can also be turned in by hand.
It's a simple design, easy to machine with a lathe and a mill. I usually make them from stainless steel, and don't even bother hardening them.
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I don't understand why you guys are recommending taper reamers? A taper reamer is great for enlarging holes in thin stock, but in a headstock it will cut a tapered hole. A tuner bushing needs a straight hole, a counterbore of a particular diameter and depth.
A straight reamer, like I've shown above is the best way to cut a precise straight hole. A pilot on the front makes it easier and neater. But, you can also use a standard Machinist's Reamer of the right size.
You can also use a large straight drill bit, gently, turning it by hand.
If you don't have any of those tools, the safest thing is a large round file. Slowly and carefully, checking the roundness and diameter.