We study the coupled hydrodynamics between a motile slender microswimmer and a deformable interface that separates two Newtonian fluid regions. From the disturbance field generated by the swimming motion, we quantitatively characterize the interface deformation and the manner in which the coupling modifies the microswimmer translation itself. We treat the role of the swimmer type (pushers and pullers), size and model an interface that can deform due to both surface tension and bending elasticity. Our analysis reveals a strong dependence of the hydrodynamics on the swimmer orientation and position. Given the viscosities of the two fluid media, the interface properties and the swimmer type, a swimmer can either migrate toward or away from the interface depending on its configurations. When the swimmer is oriented parallel to the interface, a pusher-type swimmer is repelled from the interface at short times if it is swimming in the more viscous fluid. At long times, however, pushers are always attracted to the interface, and pullers are always repelled from it. However, swimmers oriented orthogonal to the interface exhibit a migration pattern opposite to the parallel swimmers. In consequence, a host of complex migration trajectories emerge for swimmers arbitrarily oriented to the interface. We find that confining a swimmer between a rigid boundary and a deformable interface results in regimes of attraction toward both surfaces depending on the swimmer location in the channel, irrespective viscosity ratio. The differing migration patterns are most prominent in a region of order the swimmer size from the interface, where the slender swimmer model yields a better approximation to the coupled hydrodynamics.