The binding energy of an exciton bound to a neutral donor (Do;X) in GaAs quantum-well wires is calculated variationally as a function of the wire width for different positions of the impurity inside the wire by using a two-parameter wave function. There is no artificial parameter added in our calculation. The results we have obtained show that the binding energies are closely correlated to the sizes of the wire, the impurity position, and also that their magnitudes are greater than those in the two-dimensional quantum wells compared. In addition, we also calculate the average interparticle distance as a function of the wire width. The results are discussed in detail.