Near-threshold photodetachment microscopy of mononegative ions is theoretically studied when a homogeneous transverse magnetic field B is superimposed to the static electric field E. The electron flux distribution is obtained using two different approaches: the quantum source theory with the energy-Green function evaluated by means of the stationary phase approximation and the closed-orbit theory. Here we describe the physical ideas while developing the theory in detail. Both theoretical methods yield consistent descriptions for all field intensities and angular momentum of the detached electrons we have investigated and also reproduce reported experimental results. From our calculations, we have found that the presence of the transverse magnetic field leads to a global displacement of the recorded electron flux distribution along the E$\times$B direction. Furthermore, we found that the shape of the recorded interference pattern changes from circular to elliptical as the magnetic field increases. We have theoretically characterized and quantified those effects on the interference pattern due to the presence of the transverse magnetic field.