We develop monotone iterative technique for a system of semilinear elliptic boundary value problems when the forcing function is the sum of Caratheodory functions which are nondecreasing and nonincreasing, respectively. The splitting of the forcing function leads to four different types of coupled weak upper and lower solutions. In this paper, relative to two of these coupled upper and lower solutions, we develop monotone iterative technique. We prove that the monotone sequences converge to coupled weak minimal and maximal solutions of the nonlinear elliptic systems. One can develop results for the other two types on the same lines. We further prove that the linear iterates of the monotone iterative technique converge monotonically to the unique solution of the nonlinear BVP under suitable conditions.