We prove approximate Lipschitz stability for monochromatic phaseless inverse scattering with background information in dimension d ≥ 2. Moreover, these stability estimates are given in terms of non-overdetermined and incomplete data. Related results for reconstruction from phaseless Fourier transforms are also given. Prototypes of these estimates for the phased case were given in Novikov (2013 J. Inverse Ill-Posed Problems, 21, 813-823).