Summary: Using a recent method based on the concept of the Kuratowski measure of noncompactness of a bounded set together with some new estimates of solutions, we prove the existence of a unique minimal pullback attractor for the evolutionary process associated with a nonautonomous nonlinear reaction-diffusion system in \(H^1_0\) in which the right-hand side satisfies only a certain integrability condition. In particular, we generalize a result obtained by \textit{Y. Li} and \textit{Ch. Zhong} [Appl. Math. Comput. 190, No. 2, 1020--1029 (2007; Zbl 1126.37049)], where at most an exponential growth of the right-hand side has been assumed for times going to both plus and minus infinity.