We study the structure of neutron stars in scalar-tensor theories for the non minimal coupling of the form (1 + kappa xi phi(2))R. We solve the hydrostatic equilibrium equations for two different types of scalar field potentials and three different equations of state representative of different degrees of stiffness. We obtain the mass-radius relations of the configurations and determine the allowed ranges for the term xi phi(2) at the center of the star and spatial infinity based on the measured maximum value of the mass for neutron stars and the recent constraints on the radius coming from gravitational wave observations. Thus we manage to limit the deviation of the model from general relativity. We examine the possible constraints on the parameters of the model and compare the obtained restrictions with the ones inferred from other cosmological probes that give the allowed ranges for the coupling constant only. In the case of the Higgs-like potential, we also find that the central value for the scalar field cannot be chosen arbitrarily, but it depends on the vacuum expectation value of the field. Finally, we discuss the effect of the scalar field potential on the mass and the radius of the star by comparing the results obtained for the cases considered here.