Separable convex resource allocation problem aims at finding an allocation of a discrete resource to several activities that minimizes a separable convex function representing the total cost or the total loss. In this paper, we consider the separable convex resource allocation problem with an additional constraint that the L ₁-distance between a given vector and a feasible solution is bounded by a given positive constant. We prove that the simplest separable convex resource allocation problem with the L ₁-distance constraint can be reformulated as a submodular resource allocation problem. This result implies that the problem can be solved in polynomial time by existing algorithms for the submodular resource allocation problem. We present specialized implementations of the existing algorithms and analyze their running time.