Heat transfer between two surfaces separated by a nanometre gap is important for a number of applications ranging from spaced head disk systems, scanning thermal microscopy and thermal transport in aerogels. At these separation distances, near field radiative heat transfer competes with heat transfer mediated by phonons. Here we quantity the contribution of phonon assisted heat transfer using lattice dynamics combined with ab-initio calculations. We clearly demonstrate that phonons dominate by far heat transfer for subnanometre gaps. Strikingly, we conclude that even in the situation where the gap is filled by air molecules, phonons provide the dominant energy channel between the two solids nearly in contact. Our results predict orders of magnitude enhanced phonon heat transfer compared to previous work and bring forward a methodology to analyse phonon transmission across nanoscale vacuum gaps between apolar materials.