The stability of spherical nanoclusters of methane and carbon dioxide hydrates in the environment of supercooled water has been studied by the molecular dynamics method under the isochoric and isobaric conditions. The process of system melting has been considered within a temperature range of 180-280 K and at pressures of 1, 50, and 100 atm (under isobaric conditions). It has been shown that clusters of CO2 hydrate melt at temperatures lower than clusters of CH4 hydrate do. The difference between the melting temperatures of the hydrates is about 40 K, which is explained by the higher solubility of carbon dioxide in water. The diffusion coefficients calculated for water and the gases attest to different mechanisms of melting their hydrates. The stability of the hydrates under the isochoric conditions appears to be lower than that under the isobaric conditions. For simulation under isobaric conditions, changes in the pressure and the degree of carbon dioxide filling have no effect on the position of the range of melting of hydrate nanoclusters.