It is shown that subcritical phase-separation nuclei (domains of decomposition) arising spontaneously in a supersaturated He3–He4 mixture can become centers of hydrodynamic vortex generation in the presence of a local nonzero or global vorticity of the liquid. The acceleration of the “rigid-body” vortex rotation of an incompressible fluid within a domain is due to the joint action of convective and Coriolis forces in the presence of a convergent radial flow whose velocity is linked by the continuity equation to the velocity of an ascending vertical flow, which increases with height. Such flows, which arise on account of the chemical and dynamical equilibrium between the domains of decomposition and the surrounding metastable He3–He4 mixture, compensate the escape of the light He3 component from the volume of the domain to the surface of the liquid on account of its buoyancy in the Earth’s gravity. Depending on the conditions of the decomposition of the solution inside the domains, acceleration of the vortices can occur according to an exponential law or by a scenario of nonlinear “explosive” instability. The formation and growth of such hydrodynamic vortices in decomposing He3–He4 mixtures gives rise to quantum vortices in the superfluid component and, as a consequence, leads to acceleration of the process of heterogeneous decomposition (phase separation) in comparison with homogeneous decomposition.