This is a very common question. I'll post a map of the circle below to make it easy.
Bill Harrison created this circle with apparent much thought and planning in 1960 – a circle in this region could not have been placed in a better position. And unlike almost all nearby circles it hasn't been moved by any of the subsequent compilers.
It's best features are:
• maximum coverage of public lands, minimal private lands
• nearly bisected by a single good road with access roads off of it
• minimal overlap into Mexico
• just barely including three important bodies of water – Arivaca Lake in the NW, Bear Grass Tank in the N, and Peña Blanca Lake in the SE.
• completely encompassing all of the Sycamore Canyon-California Gulch southward-flowing drainages
• and, completely unbeknownst to Bill, incorporating the entire US range of Five-striped Sparrow (the species wasn't discovered here until a few years later)
And Bill did this all without Google Earth. (He may have even used a slide rule...)
The coordinates of the center are N31.47006°, W111.17278°
Click on the image for a larger version.