The state estimation problem for discrete-time Markov jump linear systems affected by multiplicative process noises is studied. The available measurements for the system under consideration have two components, the model measurement and the output measurement, where the model measurement is affected by a fixed amount of delay. By using the Bayes theorem and some results obtained in this paper, a novel suboptimal state estimation algorithm is proposed in the sense of minimum mean-square error. The proposed algorithm is recursive and does not increase computation and storage load with time. Computer simulations are carried out to evaluate the performance of the proposed algorithm. The simulation results show the superiority of the proposed algorithm.