This paper presents a highly efficient and effective approach to bound the first excursion probability of linear stochastic FE models subjected to imprecise Gaussian excitations. In previous work, some of the authors proposed a highly efficient approach based on the operator norm framework to bound such first excursion probabilities without having to resort to double-loop problems [1]. However very efficient, the approach presented in [1] is limited to deterministic models, or models containing epistemic uncertainty. In this paper, the classic operator norm approach is augmented by linearising the stochastic FE model around the mean of the aleatory uncertain parameters. This allows for determining those values of the epistemically uncertain parameters that yield an extremum in the failure probability without solving the associated reliability problem. Hence, the double loop that is typically associated to this type of problems is effectively broken. A case study illustrates the effectiveness and efficiency of the proposed method.