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Heterostructures based on atomic monolayers are emerging as leading materials for future energy efficient and multifunctional electronics. Due to the single atom thickness of monolayers, their properties are strongly affected by interactions with the external environment. We develop a model for interface thermal conductance (ITC) in an atomic monolayer van der Waals bonded to a disordered substrate. Graphene on SiO2 is initially used in our model and contrasted against available experimental data; the model is then applied to monolayer molybdenum disulfide (MoS2) on SiO2 substrate. Our findings show the dominant carrier of heat in both graphene and MoS2 in the cross-plane direction is the flexural (ZA) phonon mode, owing to the large overlap between graphene ZA and substrate vibrational density of states. The rate of phonon transfer across the interface depends quadratically on the substrate coupling constant Ka, but this interaction also causes a lifting of the lowest flexural phonon modes. As a result, ITC depends roughly linearly on the strength of the coupling between a monolayer and its substrate. We conclude that, in both graphene and MoS2 on SiO2, substrate adhesion plays a strong role in determining ITC, requiring further study of substrate coupling in TMDCs.