The predictive map hypothesis provides a promising framework for understanding hippocampal function. A prominent formalization of this idea, the successor representation (SR), posits that each place cell encodes the expected future occupancy of its corresponding location, enabling predictive spatial coding. While recent studies have investigated biologically plausible mechanisms for learning the SR in hippocampal circuits, it remains unclear how such predictive representations may emerge from upstream entorhinal inputs, such as grid cells. In this study, we propose a biologically inspired neural model that constructs SR-like predictive maps from medial entorhinal cortex (MEC) grid cell activity. The model first generates hippocampal place cell-like representations by a recurrent network that computes nonnegative sparse coding, and then transforms them into predictive representations using a Hebbian-like learning rule. We show that the proposed model can reproduce skewed place fields, a hallmark of predictive maps. Furthermore, we demonstrate that the predictive representations can support spatial navigation in continuous environments. These results offer a simplified and biologically inspired framework for predictive map learning, bridging MEC grid cell activity and hippocampal predictive activity through local and plausible learning mechanisms.