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The understanding and modeling of the heat transport across nanometer and sub-nanometer gaps where the distinction between thermal radiation and conduction become blurred remains an open question. In this work, we present a three-dimensional atomistic simulation framework by combining the molecular dynamics (MD) and phonon non-equilibrium Green's function (NEGF) methods. The relaxed atomic configuration and interaction force constants of metallic nanogaps are generated from MD as inputs into harmonic phonon NEGF. Phonon tunneling across gold-gold and copper-copper nanogaps is quantified, and is shown to be a significant heat transport channel below gap size of 1nm. We demonstrate that lattice anharmonicity contributes to within 20-30% of phonon tunneling depending on gap size, whereas electrostatic interactions turn out to have negligible effect for the small bias voltage typically used in experimental measurements. This work provides detailed information of the heat current spectrum and interprets the recent experimental determination of thermal conductance across gold-gold nanogaps. Our study contributes to deeper insight into heat transport in the extremely near-field regime, as well as hints for the future experimental investigation.