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We investigate the growth of gas bubbles in a water solution at rest with a supersaturation level that is generally associated with diffusive mass transfer. For CO2 bubbles, it has been previously observed that, after some time of growing in a diffusive regime, a density-driven convective flow enhances the mass transfer rate into the bubble. This is due to the lower density of the gas-depleted liquid which surrounds the bubble. In this work, we report on experiments with different supersaturation values, measuring the time t(conv) it takes for convection to dominate over the diffusion-driven growth. We demonstrate that by considering buoyancy and drag forces on the depleted liquid around the bubble, we can satisfactorily predict the transition time. In fact, our analysis shows that this onset does not only depend on the supersaturation, but also on the absolute pressure, which we corroborate in experiments. Subsequently, we study how the depletion caused by the growth of successive single bubbles influences the onset of convection. Finally, we study the convection onset around diffusively growing nitrogen N-2 bubbles. As N-2 is much less soluble in water, the growth takes much longer. However, after waiting long enough and consistent with our theory, convection still occurs as for any gas-liquid combination, provided that the density of the solution sufficiently changes with the gas concentration.