There is a certain amount of confusion over the exact location of Rhosgobel. The initial draft map for The Lord of the Rings, reproduced in volume 7 of The History of Middle-earth, shows it in the far south of the forest, precariously close to Dol Guldur. However, in the notes to The Istari in Unfinished Tales, Christopher Tolkien quotes the location as being 'in the forest borders between the Carrock and the Old Road', which would place it much farther to the north.
The latter location seems to fit the established dating rather better, allowing for Radagast to remain there safely long into the Third Age. However, the only remotely canonical statement comes from The Fellowship of the Ring II 3, where the scouts of Elrond 'had come down into Wilderland and over the Gladden Fields and so at length had reached the old home of Radagast at Rhosgobel.' This description can only reasonably apply to the southern location, near Sauron's stronghold at Dol Guldur.
The relations between these two variable accounts are unclear. The southern version appears to be the older, so perhaps Tolkien intended to revise Rhosgobel's location to place it in less direct danger from Sauron. If so, however, that change never made it into the text of The Lord of the Rings. We can place Rhosgobel, therefore, on the western fringe of the Narrows of the Forest, opposite the East Bight.