If you are coming from out of town, you may want to find lodging near the circle. If your area is in the eastern half of the circle, staying near Tubac, Rio Rico, or in Nogales would be the best bet. If your area is in the western half of the circle, staying in Amado or even Green Valley would be closer – you'd then access the circle via Arivaca Road.
Of course, there is camping possible all over in the circle. Any flat spot on USFS property is fair game unless it's marked otherwise.
Here are some suggestions for lodging in the area. Feel free to add to this in the comments section.
P.O. Box 636, Amado, Arizona 85645
Amado Territory Inn
P.O. Box 81, 3001 E. Frontage Road, Amado, AZ 85645
Toll Free: (888) 398-8684
Local: (520) 398-8684
Fax: (520) 398-8186
Rio Rico Esplendor Resort
1069 Camino Caralampi, Rio Rico, AZ 85648
Toll Free: 1-800-288-4746
Best Western Sonora Inn and Suites
750 West Shell Road, Nogales, AZ 85621
Holiday Inn Express
850 West Shell Road, Nogales, AZ 85621
Candlewood Suites Nogales
875 North Frank Reed Road, Nogales, AZ 85621
Best Western Siesta Motel
673 North Grand Avenue, Nogales, AZ 85621
547 West Mariposa Road, Nogales, AZ US 85621