We calculate two-photon transition rates to exciton states in GaAs quantum wells as a function of the well width. We obtain rates for transitions to states of s symmetry as well as of p symmetry. The exciton states are obtained by variational calculations using wavefunctions which are separable in the spatial coordinates. For the calculation of the transition rates, the sum over the intermediate exciton states is taken explicitly into account. We find large transition rates for s-states, which indicate that these transitions give rise to peaks in the absorption spectrum, in agreement with experiment. The transition rates of s-states increase considerably for decreasing well widths. On the other hand we find that transitions to p-states have negligible rates, nearly independent on the well width, and we do not expect them to be observable as well-distinguishable peaks.