The pertechnetate ion Tc^VIIO₄^- is a nuclear fission product whose major issue is the high mobility in the environment. Experimentally, it is well known that Fe₃O₄ can reduce Tc^VIIO₄^- to Tc^IV species and retain such products quickly and completely, but the exact nature of the redox process and products is not completely understood. Therefore, we investigated the chemistry of Tc^VIIO₄^- and Tc^IV species at the Fe₃O₄(001) surface through a hybrid DFT functional (HSE06) method. We studied a possible initiation step of the Tc^VII reduction process. The interaction of the Tc^VIIO₄^- ion with the magnetite surface leads to the formation of a reduced Tc^VI species without any change in the Tc coordination sphere through an electron transfer that is favored by the magnetite surfaces with a higher Fe^II content. Furthermore, we explored various model structures for the immobilized Tc^IV final products. Tc^IV can be incorporated into a subsurface octahedral site or adsorbed on the surface in the form of Tc^IVO₂·xH₂O chains. We propose and discuss three model structures for the adsorbed Tc^IVO₂·2H₂O chains in terms of relative energies and simulated EXAFS spectra. Our results suggest that the periodicity of the Fe₃O₄(001) surface matches that of the TcO₂·2H₂O chains. The EXAFS analysis suggests that, in experiments, TcO₂·xH₂O chains were probably not formed as an inner-shell adsorption complex with the Fe₃O₄(001) surface.