Recent studies have revealed that Janus structures and heterobilayers made from them might have properties superior to those of two-dimensional (2D) materials. We construct 2D monolayer boron phosphide (MBP)/MoSSe and MoSSe/MBP heterobilayers and describe comprehensively their optoelectronic properties in the presence of biaxial in-plane and uniaxial out-of-plane strain and the effects of electric fields using first-principles methods. The electronic bands of both MBP/MoSSe and MoSSe/MBP heterobilayers display a peculiarity in the direct gap, with different band-gap values in the pristine forms. Remarkably, it is shown that different varieties of band alignments are induced for different orders of the van der Waals (vdW) heterobilayers. Electric fields and in-plane and out-of-plane strain give rise to important changes in the electronic and optical properties. The band alignments can transform from type I to type II under the influence of an electric field and strain. The main absorption peaks of pristine forms of both heterobilayers have in the visible region (approximately 2.67 eV), while in the presence of biaxial strain are redshifted. Since the absorption peaks of the pristine forms of the heterobilayers are in the visible region, the heterobilayers recommended here can be used in photovoltaic applications, and the presence of effects of external electric fields and strain promises functional optoelectronic devices.