The fast neutron fluence is an important parameter for the reactor pressure vessel (RPV) lifetime. The uncertainty estimation of this parameter is crucial to manage the RPV with a suitable safety margin. This work focuses on a facet of the problem that concerns the uncertainty contribution of the spectrum of the fission source for different burnups in a thermal neutron reactor. The main goal of this paper is to assess the effect of a possible uncertainty correlation among the spectra of the fissile nuclei, involved in the fission source, on the response uncertainty. Two main simplifications are assumed to reduce the complexity of the problem. The first simplification concerns the geometry of the transport problem that is chosen to calculate as fast as possible the sensitivities and the different responses. The second simplification is related to the way by which one can take into account the correlations among spectra of different fissile nuclei. Simple ENDF-6 models of the fission spectrum (Maxwell, Watt, and simplified Madland-Nix) are used to define correlations among the fissile spectra through the mean neutron energy of the prompt fission spectrum. Results are given to quantify the effect of these postulated correlations on response uncertainties and are compared to the ones using JENDL-4.0 covariances.