We determine the temperature profile in magnetic nanocontacts submitted to the very large current densities that are commonly used for spin-torque oscillator behavior. Experimentally, the quadratic current-induced increase of the resistance through Joule heating is independent of the applied temperature from 6 K to 300 K. The modeling of the experimental rate of the current-induced nucleation of a vortex under the nanocontact, assuming a thermally-activated process, is consistent with a local temperature increase between 150 K and 220 K. Simulations of heat generation and diffusion for the actual tridimensional geometry were conducted. They indicate a temperatureindependent efficiency of the heat sinking from the electrodes, combined with a localized heating source arising from a nanocontact resistance that is also essentially temperature-independent. For practical currents, we conclude that the local increase of temperature is typically 160 K and it extends 450 nm about the nanocontact. Our findings imply that taking into account the currentinduced heating at the nanoscale is essential for the understanding of magnetization dynamics in nanocontact systems.