We consider n x n hyperbolic systems of balance laws in one space dimension partial derivative H-t(u) partial derivative F-x(u) = G(u),t > 0, 0 < x < L, under the assumption that all negative (resp., positive) characteristics are linearly degenerate. We prove the local exact one-sided boundary null controllability of entropy solutions to this class of systems, which generalizes the corresponding results obtained in [T. Li and L. Yu, T. Math. Pares Appl. (9), 107 (2017), pp. 1-40] from the case without source terms to that with source terms. In order to apply the strategy used in [T. Li and L. Yu, T. Math. Pures Appl. (9), 107 (2017), pp. 1-40], we essentially modify the constructive method by introducing two different kinds of approximate solutions to the system in the forward sense and to the system in the rightward (resp., leftward) sense, respectively, while their limit solutions are equivalent to some extent.