When a soluble solid body is suddenly put in contact with water, a convection flow can be generated. Once the fluid layer charged into solute is sufficiently dense, this layer becomes unstable under the action of the buoyancy forces. We perform here a linear stability analysis in order to predict the time of appearance of the convection flow, the onset time, and the associated wavelength. As the base state evolves with time due to the solute diffusion, the usual theoretical methods cannot be used. We show that the criterion of marginal instability with a "frozen base state" used for convection in porous media fails for providing the onset parameters in fluid convection. Here, using a modified criterion, i.e., the instability growth rate must be larger than the time evolution of the base state, we find the onset parameters in satisfying agreement with the previous experimental and numerical works. Our results complete our previous numerical work [J. Philippi et al., "Solutal convection induced by dissolution," Phys. Rev. Fluids 4, 103801 (2019)] in order to determine the conditions for generating a convective flow under the action of dissolution.