We analyze the sigma meson mass and width together with the pion-pion scattering parameters in the context of nonlocal chiral quark models with wave function renormalization (WFR). We consider both nonlocal interactions based on the frequently used exponential form factor, and on fits to the quark mass and renormalization functions obtained in lattice calculations. In the case of the sigma properties, we obtain results which are less dependent on the parametrization than in the standard local Nambu-Jona-Lasinio model, and which are in reasonable agreement with the recently reported empirical values. We also show that the inclusion of the WFR tend to improve the description of the pi-pi scattering parameters, with the lattice inspired parametrization providing the best overall results. Finally, we analyze the connection of the nonlocal quark models discussed here with chiral perturbation theory, and present the model predictions for the low-energy constants relevant for pi-pi scattering to O(4) in the chiral expansion.