At low temperatures and He concentrations below -6.6%, there is experimental evidence about the existence in liquid helium mixtures of stable vortices with He-rich cores. When the system is either supersaturated or submitted to a tensile strength, vortices lose stability becoming metastable and eventually completely unstable, so that their cores freely expand. Within a density functional approach, we have determined the pressure- He concentration curve along which this instability appears at zero temperature.