In order to obtain information about the internal structure of fluctuating Cooper pairs in the pseudogap state and below the transition temperature of high T_c superconductors, we solve the Bethe-Salpeter equation for the two-electron propagator in order to calculate a “pair structure function” g_P(P,\pmbr)g_{P}({\mathbf{P}},\pmb{\rho}) that depends on the internal distance \pmbr\pmb{\rho} between the partners and on the center of mass momentum P of the pair. We use an attractive Hubbard model with a local potential for s-wave and a separable potential for d-wave symmetry. The amplitude of g_P for small ρ depends on temperature, chemical potential and interaction symmetry, but the ρ dependence itself is rather insensitive to the interaction strength. Asymptotically g_P decreases as an inverse power of ρ for weak coupling, but exponentially when a pseudogap develops for stronger interaction. Some possibilities of observing the pair structure experimentally are mentioned.