In this work, the impact of different HfO2/Al2O3-based multilayer dielectric stack (DS) configurations on the electrical characteristics and on the resistive switching (RS) performance of Ni/Insulator/Silicon devices has been systematically investigated. Significant differences are observed in the electrical characteristics of the fabricated bilayer, trilayer and pentalayer stacks compared to a single HfO2 layer of the same physical thickness. The RS analysis has shown similar low resistance state currents and set voltages for all the DS combinations whereas currents at the high resistance state and reset voltages depend on the DS. The shift of the reset voltage to lower values for HfO2 and Al2O3/HfO2/Al2O3 cases is explained by the results from thermal simulations that reveal that these differences could be associated to the different temperature distributions at the narrowest part of the conductive filament immediately before the thermally triggered reset process occurs.