The reservoir area dependent thermal transport at nanoscale two-dimensional and one-dimensional interfaces is investigated by the non-equilibrium Green's function method. For the two-dimensional nanoscale interface composed of graphene sheets, the reservoir area is identical to the contact area S at the interface. As S increases from few atoms, the interfacial thermal conductance σ per S (Λ = σ/S) is negatively dependent on S due to the decrease of phonon transmission per S. With S increasing to several square nanometers, Λ converges to a constant value. However, for the one-dimensional nanoscale interface composed of nested carbon nanotubes (NCNTs), it is σ instead of Λ that converges to a constant value because the reservoir in one-dimensional nanoscale NCNTs has a fixed area, which can only provide finite transport channels. There are two competitive factors influencing the thermal transport at the interface in the NCNT model. One is phonon mode coupling and the other is phonon scattering. These two factors lead to an interesting trend of σ that as the overlap between NCNTs increases, σ increases at first and then decreases and converges to a constant value. These findings indicate that the thermal transport behavior has a strong dependence on the contact details and reservoir area at the nanoscale interface.