This paper studies the existence of aperiodic solution of a nonlinear integrodifferential system of the form x′(t)=Dx(t)+f(x(t))+t∫−∞k(t,s)g(x(s))ds+p(t), for each continuous periodic function p and under suitable assumptions on f, k and g. A topological transversality method is employed to obtain the existence of periodic solutions. This method relies on a priori bounds on periodic solutions. Several examples are provided where a variant of Liapunov's direct method is employed to obtain a priori bounds on periodic solutions.