Motivated by (Xu et al. in Bound. Value Probl. 2013:262, 2013) and (Yang and Ren in Proc. Indian Acad. Sci. Math. Sci. 124(2):175-178, 2014), in this paper we aim to construct a modified Green function in the upper-half space of the n-dimensional Euclidean space, which generalizes the boundary property of general Green potential.