This paper reports a structural study of 3 and 10 monolayer (ML) thick NiO films grown on Ag(001). Polarization-dependent X-ray absorption spectroscopy at the Ni K edge allowed us to obtain an accurate description of the local atomic environment of the Ni atoms up to the seventh coordination shell, including the determination of the in-plane and out-of-plane strains for the NiO films (determined with an accuracy better than 1%, resulting from the determination of Ni-O bond distances with a statistical error lower than 0.01 Angstrom). Multishell multiple scattering calculations have been used to fit the data. Within the experimental errors, the 3 ML film has both in and out-of-plane Ni-O distances compatible with those expected in the case of perfect pseudomorphism on Ag(001). A rough film morphology, possibly including also the presence of NiO islands, is also suggested by the optimized coordination numbers. Conversely the 10 ML film is partially relaxed without significant deviation from ideal bidimensional film. The present EXAFS study allows us to rule out any significant atomic interdiffusion process between the substrate and the NiO film. Comparison with previous literature data allows us to make the hypothesis of a two step growth mechanism for NiO films on Ag(001). The NiO-Ag(001) interface distance determined in this EXAFS study (d(interface) = 2.36 +/- 0.05 Angstrom) is in remarkable agreement with theoretical calculations (d(interface) = 2.38 Angstrom).