In this paper, we use the Eigenfunction method (Ritus'method) to calculate the Klein-Gordon propagator under an external magnetic field with an exponential damping factor b, in a four-dimensional Euclidean space. We write the propagator of scalar field in terms of plane waves and Laguerre polynomials. Also the eigenvalues associated to differential operator show the quantization in the xy-plane through the Landau levels. We apply the calculated propagator to the mass parameter of an interacting boson system under an exponentially decaying magnetic field.