The delivery of enriched icy grains has been proposed to explain the enrichment of Jupiter with noble gases. While consideration has been given to the formation composition of the ices, the release of species as grains migrates inward toward the forming planets has not been studied extensively. The accretion and subsequent desorption of noble gas species alongside water into an enriched icy grain has been simulated with a Monte Carlo approach. Experimental measurements of desorption rates have been used to constrain the desorption rates of each noble gas species. Our results show consistency with the Galileo measurements of Jupiter. There is an indication that if the noble gases were delivered in thin ices, the Ar/Xe atmospheric signature on Neptune could be <1 rather than ~1.3 observed on Jupiter. The delivery of 0.1 to 0.5 Earth masses of enriched ice appears to be sufficient to provide the measured Jovian enrichment.