Summary: We consider the monotone variational inequality of finding \(x^*\in C\) such that \(\langle(I-T)x^*, x-x^*\rangle\geq 0\) for \(x\in C\), where \(C\) is a closed convex subset of a real Hilbert space and \(T\) is a nonexpansive self-mapping of \(C\). Techniques of nonexpansive mappings are applied to regularize this variational inequality. The regularized solutions and an iteration process are shown to converge in norm to a solution of this variational inequality.