We propose a new stochastic variance-reduced damped L-BFGS algorithm, where we leverage estimates of bounds on the largest and smallest eigenvalues of the Hessian approximation to balance its quality and conditioning. Our algorithm, VARCHEN, draws from previous work that proposed a novel stochastic damped L-BFGS algorithm called SdLBFGS. We establish almost sure convergence to a stationary point and a complexity bound. We empirically demonstrate that VARCHEN is more robust than SdLBFGS-VR and SVRG on a modified DavidNet problem - a highly nonconvex and ill-conditioned problem that arises in the context of deep learning, and their performance is comparable on a logistic regression problem and a nonconvex support-vector machine problem.