van der Waals heterostructures (vdWHs) open the possibility of creating novel semiconductor materials at the atomic scale that demonstrate totally new physics and enable unique functionalities, and have therefore attracted great interest in the fields of advanced electronic and optoelectronic devices. However, the interactions between metals and vdWHs semiconductors require further investigation as they directly affect or limit the advancement of high-performance electronic devices. Here we study the contact behavior of MoS₂/WSe₂ vdWHs in contact with a series of bulk metals using ab initio electronic structure calculations and quantum transport simulations. Our study shows that dual transmission paths for electrons and holes exist at the metal–MoS₂/WSe₂ hetero-bilayer interfaces. In addition, the metal-induced bandgap state (MIGS) of the original monolayer disappears due to the creation of the heterolayer, which weakens the Fermi level pinning (FLP) effect. We also find that the creation of the heterolayer causes a change in the Schottky barrier height (SBH) of the non-ohmic contact systems, whilst this does not occur so easily in the ohmic contact systems. In addition, our results indicate that when Al, Ag and Au are in contact with a MoS₂/WSe₂ hetero-bilayer semiconductor, a low contact barrier exists throughout the whole transmission process causing the charge to tunnel to the MoS₂ layer, irrespective of whether the MoS₂ is in contact with the metals as the nearest layer or as the next-nearest layer. Our work not only offers new insights into electrical contact issues between metals and hetero-bilayer semiconductors, but also provides guidance for the design of high-performance vdWHs semiconductor devices.