The existence of “radial” conformai Killing (RCK) vector fields is discussed for metrics describing spherically symmetric, shear-free, perfect fluids. It is shown that conformally flat metrics admit three RCK fields, while nonconformally flat metrics might admit, at most, one such field. An RCK vector parallel to the 4-velocity of the fluid occurs in a subclass of conformally flat metrics containing the Friedmann-Robertson-Walker space-times as particular cases. A new class of nonconformally flat, self-similarity solutions is found. The necessary conditions for the existence of an RCK field in other nonconformally flat metrics are given in full.