The binding energies of hydrogenic impurities in both infinite and finite GaAs-(Ga,Al)As cylindrical quantum-well wires are calculated as functions of the wire radius and of the impurity location in the well for different radii of the wires using a variational procedure within the effective-mass approximation. Assuming there is no intentional doping, we treat the impurity position as a random variable and define a density of impurity states that we have calculated as a function of the impurity binding energy. As a general feature, the density of impurity states presents two structures associated with impurities at the center and at the edge of the quantum-well wire that may be important in the understanding of absorption and photoluminescence experiments of doped GaAs-(Ga,Al)As quantum-well wires.