The van der Waals (vdW) heterostructures are emerging as promising structures for future possible optoelectronic devices. Motivated by the recent studies on vdW heterostructures with their fascinating physical properties, we investigate the electronic and optical properties of boron phosphide/blue phosphorus heterostructures in the framework of density functional theory (DFT) and tight-binding (TB) approximations. We analyze the variation of the energy band gap, the characteristics of the energy band diagram, charge redistribution by stacking and the electrostatic potential along the perpendicular direction. The dynamical stability of these structures is ensured by the phonon spectra. We show that trilayer heterostructures of boron phosphide/bilayer blue phosphorus are in-direct band gap semiconductors while heterobilayers have a direct band gap at the K point. Moreover, we examine the optical properties of monolayer boron phosphide and heterostructures as part of DFT calculations. We conclude that the heterostructures have remarkable optical absorption over the UV range together with being transparent to the visible spectrum, and may be a prominent material for future optoelectronic devices.