This work discusses the existence of weak solutions for a system of Kirchhoff-type involving variable exponent (α1(m), α2(m))-Laplacian operators and under the Dirichlet boundary conditions. Under appropriate hypotheses on the nonlinear terms and the Kirchhoff functions, the existence of weak solutions is obtained on the spaces W1,α1(m) 0 (D) × W1,α2(m) 0 (D). The proof of the main result is based on a topological degree argument for a class of demicontinuous operators of (S+)-type.