This paper presents a framework for probability sensitivity estimation of a class of problems involving linear stochastic finite element models. The sensitivity measure consists of the derivative of the failure probability with respect to the statistics of the underlying random field associated with the model. The framework is formulated as a post-processing step of Line Sampling and it is implemented considering two different approaches. The performance of these two approaches is studied by means of numerical examples. It is concluded that both offer effective means for estimating the sought sensitivity measure. Furthermore, it is observed that the correlation length associated with the random field controls the magnitude of the sensitivity measure.