The scarcity and high cost of raw materials such as platinum to synthesize catalysts, makes it desirable to reduce the amount of precious metal used preserving their overall catalytic performance. A promising solution to this is for example, using highly dispersed and catalytically active transition metal clusters supported on metal oxides, such as TiO₂, CeO₂, and γ-Al₂O₃. Thus, bimetallic clusters can potentially offer additional catalytic enhancement due to a synergy between the involved metal atoms. We focus our attention on a computational, DFT study of small bimetallic Re-Pt clusters supported on γ-Al₂O₃. We aim to determine global minima configurations of Re-Pt clusters (up to 5 atoms) on the γ-Al₂O₃ (100) surface by using a Basin Hopping global optimization scheme. The adsorption and mixing energies are calculated as a function of clusters size and composition, thus providing a quantifiable picture of the stability of such bimetallic clusters. Our results show that bimetallic Re-Pt supported clusters exhibit higher stability compared to monometallic Re or Pt clusters. Regarding the bonding mechanism between metal clusters and the oxide substrate, our calculations indicate that the contribution to the PDOS of the 6-s states of both Re and Pt atoms is negligible, compared to that of the 5s-states. Furthermore, projected density of states (PDOS) analysis indicate that no bond is formed between the surface Al atoms and clusters. A Bader charge transfer analysis reveals that Re-rich clusters transfer electrons to the γ-Al₂O₃ substrate, while the opposite is true for their Pt-rich counterparts. Finally, various cluster nucleation pathways are investigated finding that the most favorable nucleation routes lead to the Re₃Pt₂ and Re₅ clusters, while the coalescence of Pt atoms into Pt clusters is energetically unfavorable.